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Mathematics Examination — 563-212

Secondary Cycle One Year One June 2009

Montréal Canadiens 100Montréal Canadiens 100thth AnniversaryAnniversary

Administration and Marking Guide

Secondary Cycle One Year One – 563-212 Administration and Marking Guide Montréal Canadiens 100th Anniversary Page 1

Administration and Marking Guide

Design Team

Teachers: Grant, Sandra N.F.S.B. Muzzerall, Erica St. Georges School. Rouette, Julie R.S.B. Wallace, Melissa N.F.S.B.

Coordination

Joanne Malowany, Math Consultant, New Frontiers S.B

Editing

Patricia Juliano, BIM, Société GRICS

Layout and Computerization

Martine Sanscartier, BIM, Société GRICS

Table of Contents 1. Presentation of the examination ........................................................................................................... 3

1.1 Description of the materials ........................................................................................................... 3 1.2 Description of the evaluation situations and connections to the Québec

Education Program (QEP) ........................................................................................................... 4-5 2. Suggested time frame for administering the different parts of the examination ..................................... 6 3. Possible adjustments ............................................................................................................................ 6 4. Conducting the activities related to the situational problem ................................................................... 7

4.1 Initial preparation ........................................................................................................................... 7 4.2 Setting the context ......................................................................................................................... 7 4.3 General procedure ........................................................................................................................... 8 4.4 Integration ...................................................................................................................................... 9

5. Conducting the activities related to Part 2

Uses mathematical reasoning and Communicates by using mathematical language ........................ 10 5.1 Initial preparation ......................................................................................................................... 10 5.2 General procedure ....................................................................................................................... 10

6. Using the results ................................................................................................................................. 11 7. The record of learning and evaluation as a basis for the end-of-year report ....................................... 11 8. Explanation of Marking Procedures ..................................................................................................... 12 9. Student profile ...................................................................................................................................... 12 Part 1 10. Marking students’ work on the situational problem .............................................................................. 13

10.1 Example of a correct solution 10.2 Observable elements related to the task


Part 2

11. Marking the Competency 2 and Competency 3 situations .......................................................... 20 Appendices

Appendix A1 – Competency 1 evaluation chart .....................................................................................



Appendix B – Guidelines Regarding Assistance Provided ..................................................................

Appendix C – Record Sheet: Help Provided ........................................................................................

Appendix D1 – Descriptive Evaluation Chart for the competency: Solves a situational problem ..........

Appendix D2 – Descriptive Evaluation Chart for the competency: Uses mathematical reasoning .......

Appendix D3 – Descriptive Evaluation Chart for the competency: Communicates by using mathematical language ................................................................................................

Appendix E – Overall Assessment of Student’s Work on the Examination ........................................


1. Presentation of the examination This examination is consistent with the principles regarding the evaluation of learning outlined by the Ministère de l'Éducation, du Loisir et du Sport. It was developed in conjunction with education consultants and teachers from various school boards in Québec. This examination consists of two parts. The first part involves a situational problem intended to evaluate the competency Solves a situational problem. The second part is made up of 10 situations that focus on two competencies: Uses mathematical reasoning and Communicates by using mathematical language. The evaluation situations in this examination focus on the main concepts and processes covered in year one of the Secondary Cycle One Mathematics program. This guide provides information about evaluating student work on the situations that make up the examination. It also includes sample answers, observable elements, and descriptive charts for each competency (Appendices D1, D2, and D3). Given that this is a mid-cycle assessment tool, educational institutions and teachers can incorporate all or part of this examination into their schedules of educational activities. This exam can serve as an information and professional development tool for teachers working with students in Secondary Cycle One. Assessment situations not used in the examination can be added to the teacher's bank of situations or used as models for developing similar types of problems. 1.1 Description of the materials

The following documents are provided for the students:

Situation focusing on competency 1 Solves a situational problem

Situations focusing on competencies 2 and 3 Uses mathematical reasoning Communicates by using mathematical language

• 1 Context document • 1 Student Answer Booklet • 1 Student Booklet

An Administration and Marking Guide is provided for teachers.


1.2 Description of the evaluation situations and connections to the Québec

Education Program (QEP)

Part I: Description of the situational problem The Montréal Canadiens 100th Anniversary

In the situational problem, the students are asked to determine the amount of money that can be raised for the Saku Koivu Foundation by engaging their school community in preparations for a charity hockey game. They will have to maximize the amount of money they can raise for this worthy cause. The following table presents concepts and processes that are likely to be used to solve the situational problem.

CONCEPTS AND PROCESSES

Arithmetic

Number sense with regard to decimal and fractional notation and operation sense Ä Fractional and decimal notation; percentage Ä Properties of operations Ä Distributive property of multiplication over addition or subtraction and

factoring out the common factor Operations involving numbers written in decimal and fractional notation Ä Estimating and rounding numbers in different situations Ä Computations: the four operations, especially with numbers written in

decimal notation, using equivalent ways of writing numbers and the properties of operations

Ä Use of a calculator: operations and sequences of operations performed in the proper order

Understanding proportionality Ä Ratios and equivalent rates Ä Ratio and proportionality coefficient

Geometry

Geometric figures and spatial sense: Ä Triangles, quadrilaterals and convex polygons Ä Base, height, area Ä Measurement: length Ä Congruent and similar figures Ä Geometric constructions Ä Geometric transformations ♦ Translation, reflection Ä Unknown measure of a segment Ä Area of polygons that can be decomposed into quadrilaterals


Part II: Description of the situations focusing on competencies 2 and 3 For each situation, the table below gives a brief description of the task to be carried out, the competency it targets and the concepts and processes involved.

Title of the situation and description of the task Competency Concepts and processes

1 .Newspaper Contest Explain how an order of operations is performed.

3 Ä Order of operations Ä Rules for signs of numbers written in decimal notation

2. Fundraising Banner Determine perimeter of letter K in banner

2 Ä Determining lengths, perimeter Ä Plane figures Ä Congruent figures Ä Relationship between SI units of length

3. Parking Lots Determine which of two parking lots offers the better deal.

2 Ä Modes of representation (algebraic expressions)

Ä Decimal calculations 4. He shoots! He Misses! Determine the measure of the angle at which player must shoot the puck. 2

Ä Unknown measurements Ä Angles: complementary, supplementary, Ä Angles formed by a transversal intersecting two parallel

lines Ä Determining measures of angles

5. Hockey Scout Create a broken-line graph and compare with data presented in a table to arrive at a conclusion.

3 Ä Drawing a graph Ä Reading graphs and tables Ä Processing data from statistical reports

6. Charity Hockey Game Determine the probability of winning a hat on first draw and pair of tickets on second draw.

2 Ä Random experiment, without replacement or order Ä Calculating the probability of an event

7. Best Ticket Seller Make a conjecture about comparing fractions whose numerators are the same.

2

Ä Comparing fractions

8. Career Rating Compare the overall +/- rating of a hockey player with the +/- rating of his first season.

2 Ä Rules of signs for numbers (integer addition)

9. Prize Giveaway Determine the number of fans among 1000 who will win all 3 prizes provided.

2 Ä Patterns Ä Properties: multiples

10. Find the Coin and Win the Prize! Determine the location of a prize in a Cartesian plane, given the coordinates of 4 points.

3

Ä Locating ordered pairs in a Cartesian plane


2. Suggested time frame for administering the different parts of the examination

Students have 3 hours to work on the situational problem (Part 1). The work can be carried over several periods. Part 2 is allotted 2 ½ hours, which can be broken into two 75-minute periods.

3. Possible adjustments1

Adjustments are suggested for this examination, but none of them involves changing its content. In fact, any change in the content of the examination, such as removing or changing a requirement, would compromise the validity of the examination.2 The suggested measures in the table below are meant for students with learning difficulties or social maladjustments, or for students who have temporary limitations because of illness or special circumstances. The school could make these adjustments for students who require special measures. It should be noted that adjustments must always be made in order to allow a student to demonstrate his or her competency, but must in no way compromise the validity of the examination. In other words, the adjustments should consist of measures related to the administration of the examination, its format, or the way in which the students may submit their work.

Possible adjustments to an end-of-year evaluation situation in mathematics

Adjustments to the environment

Ä Make a cardboard cubicle so that the student can work without distraction. Ä Use a room other than the classroom. Ä Use special lighting. Ä Use a tape recorder, headphones, and a recording of the evaluation situations. Ä Use a tape recorder to record students’ work. Ä Have an attendant read the situations aloud and reread certain instructions at a

student’s request. Ä Have an attendant write down the words and sentences that a student dictates or

make sure that the student writes legibly. Adjustments to the time allotted for the exam

Ä Allot up to 30 extra minutes for the situational problem and up to 30 extra minutes for the situations in Part 2.

Ä Schedule frequent breaks at regular intervals. Adjustments to the presentation of the exam

Ä Space out the text on the page to make it easier to read. Ä Print the exam on coloured paper (beige or blue). Ä Change the orientation of the paper on which the exam is printed (portrait, landscape) Ä Add lines to show where the student must write.

In general, the teacher may support students who feel particularly insecure by encouraging them, frequently providing them with positive feedback on their attitude, or reminding them of the fact that they have been able to do similar or related work in the past.

1 The information in this section is based on work in progress being carried out by Doris Tremblay for the MELS.

2 An instrument’s ability to measure what it is designed to measure. (Renald Legendre, Dictionnaire actuel de l’éducation, Montréal, Guérin, 2005, p. 1436.


4. Conducting the activities related to the situational problem

4.1 Initial preparation

The week before administering the situational problem • Ask the students to prepare a personal memory aid. The memory aid consists of one letter-sized

sheet of paper (8½ x 11). Both sides of the sheet may be used. Any mechanical reproduction of this memory aid is forbidden.

• Review the evaluation criteria with the students and make sure they understand them.

(Appendix A1)

4.2 Setting the Context

Introduction

To engage students in the ES, which has a hockey theme based on the 100th anniversary of the Montreal Canadiens, discuss the following:

1. Ask students to name their favourite hockey team, and to explain why they picked that team. 2. Talk about the celebration of the 100th anniversary of the Montreal Canadiens. Elicit names of any

past famous players (such as Maurice “the Rocket” Richard). 3. Ask students to indicate their favourite hockey player(s), and tell why they picked them. 4. Ask students what they know about Saku Koivu, and give them the following information.

Saku Koivu is a professional ice hockey player, who currently plays center for the Montréal Canadiens as the team captain.

On September 6, 2001, Koivu was diagnosed with Non-Hodgkin's lymphoma after experiencing severe stomach pains. Later it was discovered that Koivu had cancer. During this rough time, Koivu received vast numbers of get-well e-mails and letters. He was also in touch with Mario Lemieux, John Cullen, and Lance Armstrong, all athletes who had beaten cancer and returned to the top.

Koivu was expected to be out for the season but made a remarkable comeback for the end of the season. Canadiens fans gave Koivu a warm welcome when he skated onto the Molson Center ice on April 9, 2002, receiving an eight-minute standing ovation in his first match after he had beat the cancer. (http://en.wikipedia.org/wiki/Saku_Koivu) Because of his experience, Koivu founded the Saku Koivu Foundation. In collaboration with the Montreal General Hospital Foundation, the foundation is committed to raising money in support of Cancer & Trauma initiatives at the Montreal General Hospital of the MUHC. Depending on the amount raised by the foundation, various needed machines/services can be provided.


Potential sources of confusion for students: • In the situational problem, reference is made to advertising spaces behind the hockey nets. These

spaces are in the shape of a rectangle and are divided into two congruent rectangles for advertising. Since there are two nets, four advertising spaces are for rent. Below is a diagram that can be used to explain this concept:

• Students may also have difficulty understanding the term “revenue sources”. Explain that revenue, in

this case, is the total sales from goods and services, before expenses. Revenue sources are the activities that will generate or provide revenue.

• Students may also need an explanation of the term “expenses”. Explain that this is the amount of

money needed to pay for an item or service, or for a category of costs. For a tenant, rent is an expense. For students or parents, tuition is an expense. Food, clothing, furniture, transportation are all categories of expenses.

• Event staging refers to all the details and activities that must be considered to ensure that an event

occurs without glitches.

4.3 General procedure

Materials for each student • Context • Student Answer Booklet • Calculator (with or without a graphic display) • Geometry set (ruler, compass, protractor, etc.) • Memory aid

Each of the solid lines represent the area available for advertisement. Each one is divided into two rectangles.


The day the situational problem is administered

• Hand out copies of the Context and the Student Answer Booklet. Ask the students to go through their documents to become familiar with all of the information and requirements. You may read the description of the situation and the task, making sure that students understand what is expected and where they are to record their results. Draw the students’ attention to the evaluation criteria listed on page 1 of the Student Answer Booklet.

• Describe the basic rules for solving the problem:

− Each student will solve the situational problem on his/her own. − Students may use a calculator, but must indicate the sequence of operations involved

without, however, rewriting the detailed calculations performed with the calculator. Teachers should model how to record this sequence: for example by writing a number sentence such as 5962 x 3 = 17 886.

− Students have approximately 3 hours to solve the problem. • When time is up, collect the examination booklets. • While the students are working, assist those who need help and record the type of assistance

provided by making a note to that effect in the chart entitled Record Sheet: Help Provided (Appendix C) or in the student's Answer Booklet. If necessary, refer to Appendix B (Guidelines Regarding Assistance Provided) which lists examples of assistance that can be given to students who need help. It indicates whether different types of assistance change the content of the examination.

4.4 Integration

If possible, review the task by asking the students questions about what they found easy or more difficult, the procedure they chose to solve the problem and how they overcame their difficulties.


5. Conducting the activities related to Part 2, focusing on the competencies: Uses mathematical reasoning and Communicates by using mathematical language

5.1 Initial preparation

The week before administering Part 2

• Ask the students to review the memory aid that they prepared for the situational problem. The memory aid consists of one letter-sized sheet of paper (8½ x 11). Both sides of the sheet may be used. Any mechanical reproduction of this memory aid is forbidden. Students may supplement the memory aid they drew up for the situational problem in Part 1 of the examination.

• Review the evaluation criteria with the students and make sure they understand them.

(Appendices A2 and A3) • Remind students that any calculations they must show or justification they must provide will be

taken into account in evaluating their answers.

5.2 General procedure

Materials for each student • Student Booklet (Competency 2 and Competency 3 Situations) • Calculator (with or without a graphic display • Geometry set (ruler, compass, protractor, etc.) • Memory aid

Administering the tasks in Part 2 • On the day Part 2 is administered, ask students to go through their booklets to become familiar

with the content. Make sure they know where they must write their answers, calculations, or explanations.

• Ask them to read page 2, which lists the evaluation criteria that will be used to evaluate the

competencies associated with the different tasks presented in this part of the examination.

• Describe the basic rules for this part of the examination − Each student works alone. − Students may use a calculator, but must clearly indicate the sequence of operations involved

without, however, rewriting all the detailed calculations performed with the calculator. – Students may use resources such as a dictionary or a memory aid that they will have

prepared on their own

– The situations in the Student Booklet should be completed in approximately 2 21 hours.

• When time is up, collect the examination booklets.

• While the students are working, assist those who need help and record the type of assistance provided by making a note to that effect in the chart entitled Record Sheet: Help Provided (Appendix C) or in the student's Answer Booklet. If necessary, refer to Appendix B (Guidelines Regarding Assistance Provided) which lists examples of assistance that can be given to students who need help. It indicates whether different types of assistance change the content of the examination.


6. Using the results

The student's results on this examination may be taken into account when preparing the end-of-year report. Teachers could use the information obtained from this examination along with the information already collected during the school year to draw up the end-of-year report for mathematics. In such cases, the procedure outlined below should be followed.

7. The record of learning and evaluation as a basis for the end-of-year report

The procedure outlined below should be used to make a judgment about the student's level of development of mathematical competencies by taking into account the information collected during the administration of the examination.

♦ Before administering the evaluation situations in the examination, make a preliminary judgment regarding each student's level of competency development by analyzing a sufficient variety of student work in various contexts that call for the use of their competencies. Refer to the scales of competency levels to determine which level corresponds to the student's overall competency development.

♦ Administer the evaluation situations in the examination and analyze the student’s work and the

observations made. ♦ Compare the information gathered on the level of the student's competency development during

the learning process with the information gathered following the examination. If there are significant differences between the results gathered during the learning process and those gathered after the examination, try to determine why this is so. Below are examples of questions that could help teachers to understand these differences.

♦ Are the students familiar with the characteristics of these evaluation situations? ♦ Do the tasks involved in answering the exam questions (e.g. reading informational materials,

referring to data compiled in tables, presenting an argument using calculations, explaining one’s reasoning) resemble the ones used in class?

Finally, make a judgment regarding the level of competency development by referring to all the information gathered throughout the year, including the information resulting from this examination, if applicable..


8. Explanation of Marking Procedures

Examples of appropriate solutions are presented for each task in the examination. The scorer must exercise his or her judgment and accept any other appropriate solution. Student solutions for the examination tasks are evaluated by comparing them with the different performance levels that take into account the evaluation criteria in the Québec Education Program. The descriptive charts for each competency (Appendixes D1, D2, and D3) can be used to interpret the students' work. The five performance levels in these charts (5, 4, 3, 2, 1), which are presented as brief descriptions, make it possible to evaluate student work according to the criteria selected. Note: Level 3 is the minimum requirement for success. More information about the specific requirements of the tasks is provided for each evaluation criterion involved (see the observable elements related to each task). An overall judgment regarding a student's work on a task may be made by taking into account his/her performance level for each evaluation criterion associated with the task.

9. Student Profile

Drawing up an overall assessment of the student's work related to the different examination tasks will provide teachers with an overview of his/her results on the examination (Appendix E). This will allow teachers to see the relationship between the competencies in the Mathematics program and the examination tasks.


PART 1 10. Marking Students’ Work on the Situational

Problem Focusing on the Competency Solves A Situational Problem

Situational Problem – Montréal Canadiens 100th Anniversary

10.1 Example of an advanced solution

Seating Capacity per section

Red:

75 × 21 000 = 15 000 seats

White:

71 × 21 000 = 3000 seats

Grey:

212 × 21 000 = 2000 seats

Boxes:

211 × 21 000 = 1000 seats

Seats per age group

Red White Grey Boxes Children 0.5 × 15 000 = 7500 Adults

3013 × 15 000 = 6500

Seniors

151 × 15 000 = 1000

Children

31 × 3000 = 1000

Adults 50% × 3000 = 1500 Seniors

61 × 3000 = 500

Children 0 Adults

54 × 2000 = 1600

Seniors

51 × 2000 = 400

1 box = 40 people x boxes = 1000 ÷ 40 = 25

Seating Capacity

Total Red White Grey Boxes

15 000 3000 2000 1000

Seating breakdown per section

Children 7500 1000 None

25 boxes Adults 6500 1500 1600

Seniors 1000 500 400


Ticket proceeds per age group

Red White

Children 7500 × $25,00 = $187,500 Adults 6500 × $31.50 = $204,750 Seniors 1000 × $28.25 = $28,250 Total Red $187,500 + $204,750 + $28,250 = $420,500

Children 1000 × $12.50 = $12,500 Adults 1500 × $15.00 = $22,500 Seniors 500 × $13.75 = $6,875 Total White $12,500 + $22,500 + $6,875 = $41,875

Grey Boxes

Children 0 Adults 1600 × $9.50 = $15,200 Seniors 400 × $5.75 = $2,300 Total Grey $15,200 + $2,300 = $17,500

25 × $1800 = $45,000

Total ticket proceeds $420,500 + $41,875 + $17,500 + $45,000 = $524,875 Staging expenses Employees: 25% of $524,875 = $131,218.75 Arena: 35% of $524,875 = $183,706.25


Advertisement sales Scale: 3 cm = 1 m Note: The diagrams below have been drawn as closely to scale as possible. These are only suggested diagrams. In order to maximize profits, students should draw the largest figures possible.

Trapezoid Parallelogram

8.7 cm (2.9 m)

8.7 cm (2.9 m)

8.7 cm (2.9 m)

9 cm (3 m)

3 cm

(1 m

)

3 cm

(1 m

)

3 cm

(1 m

)

3 cm

(1m

)

Isosceles Triangle

Rectangle

9 cm (3 m)

9 cm (3 m) 3

cm (1

m)

3 cm

(1m

)

9 cm (3 m) 3 cm

(1m

)

5.4 cm 5.4 cm

Money raised from advertisements

Shapes Area (m2) Amount raised

Parallelogram 2.9

10.35 x $2000

$207,000

Trapezoid 2.95

Triangle 1.5

Rectangle 3

Total 10.35 Note: Areas and amounts raised for the required shapes will vary, depending on the dimensions of

the shapes constructed by the students. .


Diagrams may differ, depending on the tools students used for the construction.


Dividing the logo stylized K into two triangles is the most cost effective. All calculations have been done for this option. However, the logo painting expenses for the two other options are also shown.Subsequent changes in the total amount raised by the Sakui Koivu Foundation have not been calculated.

Cost of painting one stylized K Number of triangles: 2 Cost of painting: 2 × $850 = $1,700 Cost of painting 3 stylized Ks 3 X $1,700 = $5,100

Cost of painting one stylized K Number of triangles: 3 Cost of painting: 3 × $850 = $2,550 Cost of painting 3 stylized Ks 3 X $2,550 = $7,650

Cost of painting one stylized K Number of triangles: 2 Cost of painting: 2 × $850 = $1,700 Number of rectangle: 1 Cost of painting: 1 × $1100 = $1,100 Total cost: $2,800 Cost of painting 3 stylized Ks 3 X $2,800 = $8,400

Amount of money raised by fundraising activities

Revenue Sources Amount Expenses Amount Total Tickets *Advertisements

$524,875 $207,000

**Logo painting Staging Costs • Employees • Arena

$5,100 $131,218.75 $183,706.25

Total revenue $731,875 Total expenses $323,325.00 *Advertisement proceeds will vary based on the degree to which students maximized the dimensions of their shapes. **Logo-painting costs will vary based on the number of triangles and rectangles into which each K was divided. Amount of money raised: Revenue − Expense $731,875 − $320,025 = $405,250

Total amount donated to the Sakui Koivu Foundation $405,250**

** This amount has been determined in light of the advertisement proceeds and logo-painting costs presented in this guide as an example of an appropriate advanced solution.


10.2 Observable elements related to the task

Evaluation criteria for the competency

Solves a situational problem

Observable Elements

Montréal Canadiens 100th Anniversary

The student….

Cr1

Oral or written explanation showing

that the student understands the

situational problem

♦ Ticket Costs � Recognizes need to determine the number of seats per section � Recognizes need to determine the number of seats per age group within each

section � Recognizes need to set up a proportional relationship to determine the number

of seniors (1/5) and adults (4/5) in the Grey section � Recognizes need to determine the total number of boxes � Recognizes need to calculate ticket proceeds for each section, for each age

group � Recognizes need to calculate event staging costs

♦ Advertisements: � Takes into account the constraints for the advertising space:

§ Scale (3 cm : 1 m) § Minimum area is 1.5 m2 per advertisement § Maximum area is 3.0 m2 per advertisement § Recognizes that a quadrilateral with only one pair of parallel sides is a

trapezoid § Recognizes that a triangle with only two congruent sides is an

isosceles triangle § Recognizes that a right trapezoid whose smaller base is as close as

possible to 3 m would maximize the profit from advertisements § Recognizes that to maximize amount of funds raised, the fourth shape

needs to be a rectangle that measures 3 m × 1 m � Recognizes need to calculate the proceeds raised, based on area of each

advertisement ♦ Centre Ice Logo

� Recognizes need to perform 2 transformations (2 Ks drawn) � Recognizes need to decompose the K into triangles or triangles and rectangle � Recognizes need to calculate the cost of painting the entire logo

♦ End Result � Recognizes that the expenses (staging and logo painting) must be subtracted

from the revenue sources (ticket sales, advertisements) to arrive at the final total amount of funds raised for the charity


Evaluation criteria

Observable Elements

Montréal Canadiens 100th Anniversary

The student….

Cr2 Mobilization of mathematical

knowledge appropriate to the situational

problem

♦ Determines ticket proceeds • Determines number of seats per section (Red: 15 000; White: 3 000; Grey: 2 000;

Boxes: 1 000 people) • Determines number of seats per age group, per section (Red: children: 7 500;

adults: 6 500; seniors: 1 000; White: children: 1 000; adults: 1 500; seniors: 500; Grey: no children; adults: 1 600; seniors: 400; Boxes: 25 boxes)

• Determines Ticket proceeds per age group, per section (Red: children: $187,500; adults: $204,750; seniors $28,250; White: children: $12,500; adults: $22,500; seniors: $6,875; Grey: adults: $15,200; Seniors: $2,300; Boxes: $45,000)

• Determines total proceeds: $524,875 • Determines staging expenses

Employees 25% of proceeds: $131.218.15 Arena 25% of proceeds: $183,706.25

♦ Determines advertisement proceeds • Accurately constructs each shape to maximize funds raised • Correctly identifies each shape that would maximize funds raised (right trapezoid,

isosceles triangle, rectangle) • Calculates area of each advertisement (for example, parallelogram: 2.9 m2,

trapezoid: 2.95 m2, triangle: 1.5 m2, rectangle: 3 m2) • Calculates amount raised per advertisement: (for example, parallelogram: $58,000,

trapezoid: $59,000, triangle: $30,000, rectangle $60,000) • Determines Total amount raised: $207,000

♦ Determines logo transformation • Accurately translates the K, showing construction lines • Accurately reflects the K, showing construction lines

♦ Determines logo cost • Determines cost of painting logo (3 Ks), (for example comprising 6 triangles $5,100

♦ Determines final amount raised (for example $405, 250)

Cr3

Development of a solution (procedure

and answer) appropriate to the

situational problem

♦ Shows a detailed, organized set of calculations ♦ Shows that he/she has validated certain steps


PART 2 11. Marking the situations focusing on the competencies

Uses Mathematical Reasoning and

Communicates by Using Mathematical Language

1. Newspaper Contest

Description of the task: Explain how an order of operations is performed.

Theme: Arithmetic Concepts and processes Order of operations

Example of a correct procedure

Procedure Explanation

-0.5 × 4 + (130 ÷ 10) – 45 ÷ -9

-0.5 × 4 + 13 – 45 ÷ -9

– 2 + 13 – 45 ÷ -9 – 2 + 13 + 5 = 16

Step 1: Complete the operations in the brackets.

Step 2: Complete all multiplications and/or

divisions from left to right-

Step 3: Complete all additions and/or

subtractions from left to right

Evaluation criteria for the competency

Communicates by using mathematical language Observable elements 1. Newspaper Contest

The student…

Cr.1 Correct interpretation of a message (oral or written) using mathematical language.

• Correctly interprets a mathematical statement • Recognizes the order of operations in a

mathematical statement.

Cr.2 Correct production of a message (oral or written) using mathematical language.

• Clearly explains each step of the procedure

used in solving the skill-testing question • Answers the skill-testing question (16)


2. Fundraising Banner

Description of the task: Determine perimeter of letter K in banner

Theme: Geometry Concepts and processes ü Lengths ü Plane figures ü Congruent and similar figures ü Relationship between SI units of length

Example of a correct procedure Students need to convert to common units. In this example, the common unit is the metre.

10 cm = 0.1 m

4.2 dm = 0.42 m

0.6 m = 0.6 m

Length of congruent sides of isosceles triangle 0.6 ÷ 2 = 0.3 m Length of third side: 0.42 m Perimeter of figure 0.6 + 2(0.1) + 2(0.3) + 2(0.42) = 2.24 m Answer: Yes, I will have enough ribbon to add a border along the outside edges of the logo because I will

need only 2.24 m.


Evaluation criteria for the competency Uses mathematical reasoning

Observable elements 2. Fundraising Banner

The student…

Cr.3 Proper implementation of mathematical reasoning suited to the situation

• Recognizes that the task involves calculating the perimeter of a complex shape

• Recognizes that the measure of the congruent sides of the isosceles triangle is half the measure of the height of the rectangle

• Recognizes the need to convert all lengths to a common unit

Cr.2 Correct application of the concepts and processes appropriate to the situation

• Converts lengths to a common unit, e.g. metres

♦ 10 cm = 0.1 m ♦ 4.2 m = 0.42 m

• Determines the length of the congruent sides of the congruent isosceles triangles

♦ 0.3 m • Determines the perimeter of the K figure

♦ 2.24 m • Compares the perimeter with the length of the

ribbon

Cr.4 Proper organization of the steps in a proof suited to the situation

• Presents a clear and organized procedure • After comparing the perimeter with the length

of the ribbon, concludes that there will be enough ribbon to trim the logo

Cr.5 Correct justification of the steps in an appropriate procedure

• Shows work that justifies his/her answer.

Cr.1 Formulation of a conjecture appropriate to the situation


3. Parking Lots

Description of the task: Determine which of two parking is less expensive

Theme: Algebra Concepts and processes ü Algebraic expression ü Constructing algebraic expression


Johnny’s Parking Lot Freddy’s Parking Lot

Entry fee: $7.00 30 minutes costs $3.00 1 hour costs 2 x $3.00 = $6.00 n = number of hours T = total cost T = 7.00 + 6.00n T = 7.00 + 6.00(4) T = 7.00 + 24.00 T = 31.00 Cost to park for 4 hours is $31.00

Entry fee: $7.50 n = number of hours T = total cost T = 7.50 + 5.00n T = 7.50 + 5.00(4) T = 7.50 + 20.00 T = 27.50 Cost to park for 4 hours is $27.50

Freddy’s Parking Lot was less expensive



Observable elements 3. Parking Lots

The student…


• Recognizes need to take into account that

Johnny’s Parking lot rate is defined in 30 minute increments

• Creates appropriate algebraic expressions to determine the cost of parking at each parking lot


• Determines the cost of parking at Johnny’s

Parking Lot for 4 hours ($31) • Determines the cost of parking at Freddy’s

Parking Lot for 4 hours ($27.50)


• Shows his/her work in a clear and organized manner


• Shows work that justifies his/her answer • Determines that Freddy’s Parking Lot was

the less expensive choice



4. He Shoots! He Misses!

Description of the task: Determine the measure of the angle at which player must shoot the puck.

Theme: Geometry Concepts and processes ü Unknown measurements ü Angles

• Complementary • Supplementary • Alternate interior • Corresponding


∠ ABD must be 70°, since both blue lines are parallel, and line AB is a transversal (alternate interior angles.) Therefore, ∠ CDB is 70°, since it is also an alternate interior angle to∠ABD. ∠ ADB must be 45°, since it is complementary to ∠ ADE, 45°. Therefore, the angle at which Doug shot from segment CD in order to pass to André is 70°+45° = 115° Note: Students are not obliged to determine the measures of all the angles above. Some may find a

more direct way to arrive at the answer.



Observable elements 4. He Shoots! He Misses!

The student…


• Recognizes the need to use the properties of

angles formed by a transversal and two parallel lines

• Recognizes the need to apply the definition of complementary angles


• Determines the measures of any angles that

he/she deems necessary to find the measure of ∠ CDA and justifies each measure using geometric properties

• Determines the measure of ∠ CDA (115°)


• Presents work in a clear and organized

manner


• Shows work that justifies his/her answer



5. Hockey Scout

Description of the task: Theme: Statistics

Create a broken-line graph and compare with data presented in a table to arrive at a conclusion.

Concepts and processes ü Reading graphs and tables ü Arithmetic mean ü Processing data from statistical reports


Year

Dylan’s Mean points per game per year

Mea

n po

ints

per

gam

e

1 2 3 4 5 0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

Explanation: I looked at the mean and noticed that the mean points per game are relatively constant. (Some students might actually note that the range of mean points scored is less than 2.) In studying the table, I noticed that in year 4, Dylan played only 10 games, but his mean number of points scored was 1.8. This is relatively constant in comparison with his mean points scored per game in other years. Based on this data, I would hire Dylan.

Note: Other explanations are acceptable, provided they show that the student analyzed the statistical data to draw a conclusion.


Evaluation criteria for the competency Communicates by using mathematical language

Observable elements 5. Hockey Scout

The student…


• Recognizes need to determine appropriate scale for axes

• Recognizes need to use the mean points per game to justify his/her answer

• Recognizes that the broken-line graph is relatively constant even though Dylan played only 10 games in his fourth year


• Uses appropriate scale for axes

• Accurately graphs data

• Appropriately titles graph

• Bases his/her justification on appropriate arguments, taking into account that Dylan ‘s mean points scored in year 4 was 1.8

• Produces a clear message consisting of coherent ideas


6. Charity Hockey Game

Description of the task: Theme: Probability

Determine the probability of winning a hat on first draw and pair of tickets on second draw.

Concepts and processes ü Random experiment, without

replacement or order ü Calculating the probability of an event

Example of a correct procedure Initially 200 tickets are available. By the time you get to draw, there have already been fifty draws. Therefore, 150 tickets remain. Two pucks and three hats have been won, leaving 8 pucks, 3 hats and the grand prize. When it is your time to draw, the probability of winning a hat on the first draw is:

§ 501

1503 = or .02 or 2%

On the second draw, since one fewer ticket is available, your chance of winning the pair of hockey tickets is

1491 or .0067114 or 0.67114%.

The combined probability is:74501

1491

501 =x , or 0.00013 or 0.013%.



Observable elements Charity Hockey Game

The student…


• Recognizes that the probability of winning has

changed since the contest began because there have already been 50 draws (i.e. probability of random event without replacement)

• Recognizes that the sample space decreases by one element after the hat is drawn

• Recognizes that the rule of multiplication must be used to calculate the probability of two consecutive random events


• Determines that the sample space of the first

draw contains 150 elements.

• Determines the probability of winning a hat on

the first draw ⎟⎠⎞⎜

⎝⎛501 or ⎟

⎠⎞⎜

⎝⎛1503

• Determines the sample space of the second draw contains 149 elements.

• Determines the probability of winning the pair

of tickets on the second draw ⎟⎠⎞⎜

⎝⎛1491

• Determines the probability of both events

occurring ⎟⎠⎞⎜

⎝⎛74501


• Shows work in a clear and organized manner


• Shows work that justifies his/her answer



7. Best Ticket Seller

Description of the task: Theme : Arithmetic

Make a conjecture about comparing fractions whose numerators are the same.

Concepts and processes ü Different ways of writing and representing

numbers • Comparing • Switching from one way of writing

numbers to another, or from one type of representation to another

• Equivalent fractions

Example of a correct procedure Convert fractions to decimals to compare

85 = 0.625

145 = 0.357

75 = 0.714

165 = 0.313

The largest fraction is 0.714 or 75

My explanation If the numerators are all the same, we can compare fractions using their denominators. The smaller the denominator, the larger is the number. Since 7 is the smallest denominator, and all numerators are 5,

then 75 is the largest number. Therefore, Ezekiel has the best rate of sales.



Observable elements Best Ticket Seller

The student…


• Uses an appropriate strategy to arrive at a conjecture

• Takes the aspects of the situation into account


• Determines the student with the best rate of ticket sales

• Uses an appropriate strategy to compare fractions without doing any calculations




• Shows work that justifies his/her conclusion


• Formulates a conjecture based on the observations made with regard to the series of examples provided or on any other strategy used to arrive at a conjecture. e.g. Explains that for any set of fractions, whose numerators are identical, the fraction with the smaller(est) denominator is the larger(est) or the fraction with the larger(est) denominator is the smaller(est).


8. Career Rating

Description of the task:

Theme: Arithmetic

Compare the overall +/- rating of a hockey player with the +/- rating of his first season.

Concepts and processes ü Rules of signs for numbers

Example of a correct procedure Integers must be added to calculate Saku Koivu’s career +/- rating. The following information is taken directly from the table: Saku Koivu’s career +/- rating (-7) + 7 + 8 + (-7) + 7 + 2 + 0 + 5 + (-5) + 1 + (-21) + (-4) + 3 = - 11 Saku Koivu’s overall +/- career rating is – 11, which is worse than his 1995-96 rating of – 7. Note: Do not penalize students who recognized that some terms, such as 7 and -7, could be “cancelled” or

“neutralized”, so they do not necessarily have to be included in the calculations.



Observable elements Career Rating

The student…


• Recognizes that he/she must find the sum of the integers

• Uses an appropriate strategy to add the integers


• Determines the overall +/- career rating (-11)

• Compares overall +/- career rating (-11) with 1995-96 +/- rating (-7)

• Determines that the overall +/- career rating is worse than the 1995-96 +/- rating







9. Prize Giveaway

Description of the task:

Theme: Arithmetic

Determine the number of fans among 1000 who will win all 3 prizes provided.

Concepts and processes ü Number sense with regard to decimal

and fractional situations ü Patterns ü Properties

Example of a correct procedure Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300 Multiples of 25: 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300 Multiples of 60: 60, 120, 180, 240, 300 The first common multiple of 15, 25, and 60 is 300. Therefore, every 300th person of the first 1000 to enter the arena will win all three prizes. Multiples of 300 under 1000: 300, 600, 900 Of the first 1000 fans, three will win all three prizes Note: Do not penalize students who conclude that because 15 is a factor of 60, the first multiple of 60 that

is divisible by 25 will be the first common multiple.



Observable elements Prize Giveaway

The student…


• Recognizes the need to use multiples in the situation


• Determines the lowest common multiple for the situation (300)

• Determines the number of people who receive all three prizes based on the number of occurrences of the LCM among 1000 fans (3)







10. Find the Coin and Win the Prize!

Description of the task: Theme: Arithmetic

Determine the location of a prize in a Cartesian plane, given the coordinates of 4 points.

Concepts and processes ü Finding ordered pairs in a Cartesian

plane


(4, 7)

(8, 3)

(-6, -3)

(-2, -7)

(1, 0)

Answer: Alex will not win the prize. The coin is located at (1, 0)


Evaluation criteria for the competency Communicates by using mathematical language

Observable elements Find the Coin and Win the Prize!

The student…


• Recognizes that he/she must construct a Cartesian plane using an appropriate scale

• Recognizes that he/she must plot the four points

• Recognizes that he/she must draw the diagonals of the quadrilateral, and locate the intersection


• Draws a Cartesian plane with all four quadrants, using an appropriate scale

• Accurately plots the four points

• Accurately draws the diagonals of the quadrilateral

• Locates the intersection of the diagonals (1, 0)

• Determines that Alex is incorrect

APPENDICES

Appendix A1

Evaluation Chart

Solves A Situational Problem

Evaluation Criteria Indicators

Cr.1 Oral or written explanation that

the student understands the situational problem

� You present a solution that takes the requirements and

constraints into account. � You determine the useful information and the task to be

carried out. � You organize this information and determine the steps

involved.

Cr.2 Mobilization of

mathematical knowledge appropriate to the situational problem

� You use the concepts and processes needed to solve the

problem. � You correctly apply the chosen mathematical concepts and

processes.

Cr.3 Development of a solution

(i.e. a procedure and a final answer) appropriate to the situational problem

� You indicate the key elements in your solution. � You make sure your solution is clear. � You make an effort to validate your solution and make the

necessary adjustments.

Appendix A2

Evaluation Chart

Uses Mathematical Reasoning


Cr.3 Proper application of

mathematical reasoning suited to the situation

� You Identify all the aspects of the task. � You use effective strategies so that the steps in your

procedure lead to an appropriate solution.

Cr.2 Correct use of the concepts

and processes appropriate to the situation

� You choose the concepts and processes appropriate to the

task. � You use the chosen mathematical processes correctly. � You use different types of representations to process the

information in an organized manner.

Cr.4 Proper organization of the

steps in an appropriate procedure

� You present a clear and organized procedure. � You clearly show your key ideas and the key steps in your

reasoning.

Cr.5 Correct justification of the

steps in an appropriate procedure

� You use appropriate mathematical arguments (laws, rules,

properties) to justify, explain, or convince.

Cr.1 Formulation of a conjecture

appropriate to the situation

� You draw conclusions or formulate a well-thought out

assumption.

Appendix A3

Evaluation Grid

Communicates By Using Mathematical Language


Cr.1 Correct interpretation of a

message involving at least one type of mathematical representation suited to the situation)

� You Identify the subject of a message by taking into account

the main elements involved in the task. � You extract correct information after processing the data. � You distinguish between the everyday and mathematical

meanings of various terms.

Cr.2 Production of a message

suited to the context, using appropriate mathematical terminology and following mathematical rules and conventions

� You define the subject and the purpose of the message. � You choose the mathematical concepts and processes that

are appropriate to the audience and to the purpose of the message.

� You base your statements on appropriate arguments. � You organize your ideas into a coherent plan.

� You follow the rules and conventions of mathematical language.

Appendix B

Guidelines Regarding Assistance Provided 3

Types of assistance that should have no bearing on the teacher’s assessment

Types of assistance to be considered in assessing the student’s work

♦ Reading and rereading the problem or part of the problem to the student

♦ Providing clarifications about the general

nature of the task ♦ Providing clarifications about everyday

vocabulary related to the context ♦ Explaining how the given information is

organized

♦ Explaining the meaning of mathematical vocabulary

♦ Highlighting or identifying useful information ♦ Breaking the task down into subtasks ♦ Providing a model ♦ Indicating concepts and processes to be

used ♦ Explaining a concept or process ♦ Rectifying the student’s work or parts of it

3. Based on the work carried out by the mathematics consultation committee in the Montérégie region.

Appendix C

Record Sheet: Help Provided

Student’s Name Type of help provided Frequency

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