Mathematics 504 CST

Mid-Year Examination

January 2011

Administration & Marking Guide

563-504

Administration Guide

Design Team: EMSB

Introduction

This examination is consistent with the principles regarding the evaluation of learning outlined by the Ministère de l'Éducation, du Loisir et du Sport. The tasks in this examination focus concepts and processes covered in the third year of the Secondary Cycle Two Mathematics program: Cultural, Scientific and Technical option (CST5). This guide provides information about the administering the evaluation situations as well as information with respect to scoring the work on the tasks that make up this evaluation situation.

1. Presentation of the examination

1.1 Description of the materials

The following documents are provided as part of this evaluation situation:

One (1) Administration and Marking Guide which contains a description of the administration conditions as well as the marking key for the student tasks.

One (1) Student Booklet for the situations focusing on Competencies 2 (Uses Mathematical Reasoning).

1.2 Description of the evaluation situations and connections to the Québec Education

Program (QEP) Number of items 16, distributed as follows

• 6 multiple choice • 4 short-constructed answers • 6 extended application answers

Description of the 6 extended answers focusing on competency 2

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 in the marking guide.

Title of the situation Concepts and processes

Question 11

A colourful display!

Solving a system of inequalities: algebraically or graphically

Question 12

An investment in stocks

Optimizing a situation, taking into account different constraints

Choosing one or more optimal solutions

Analyzing and interpreting the solution(s), depending on the context

Question 13

The Municipal Campground

Optimizing a situation, taking into account different constraints

Solving a system of inequalities: algebraically or graphically

Choosing one or more optimal solutions

Analyzing and interpreting the solution(s), depending on the context

Question 14

A visit to the museum

Representting a graph

Euler circuit

Question 15

Snow removal

Network of minimum value

Question 16

Multitasking at the airport

Critical path

2. Timetable for administering the examination and time allotted for the

evaluation situations This evaluation situation should be administered in one 3 hour time block on or after January 14th 2011.

3. Possible adjustments

Students are to do the tasks in this evaluation situation individually. No teacher assistance is permitted in Cycle Two. Teachers should consult the professionals who support the students with an Individualized Educational Plan (IEP) on a case to case basis in order to determine the appropriate adjustments for each student.

4. Procedure for administering the evaluation situation

4.1 Initial preparation

Ask the students to draw up a memory aid. Students may use a memory aid that they have prepared for another evaluation situation if it is the original hand-written copy.

Review the evaluation criteria with the students and explain the indicators for each criterion. For this purpose, you may copy the evaluation grids (Appendix A) onto transparencies.

Remind them that any required calculations or explanations will be taken into account in grading their work.

4.2 General procedure

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

5. Administering the evaluation situation:

On the day of the evaluation situation, ask students to go through their booklet to become familiar with its content. Also, make sure they know where in their booklet they must write their answers, calculations, or explanations.

Ask them to read page the instructions and the evaluation criteria that will be used to evaluate their level of competency development in the different task. Located at the bottom of the pages in the student booklet are evaluation grids which indicate the criteria to be applied in each situation. In the marking guide, you will find more information about the specific requirements of the tasks as well as interpretation tools to determine the student's performance level (1, 2, 3, 4, and 5) for each evaluation criterion involved.

Describe the basic rules: − 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.

– Student may use resources such as a dictionary or a memory aid that they will

have prepared on their own. 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.

– The booklet should be completed within the time frame indicated on the cover

page of the document. When time is up, collect the examination booklets.

6. Marking Key

PART A: Multiple-Choice Questions Questions 1 to 6 4 marks or 0 marks

D C

A A

C B

PART B: Short-Constructed Answer Questions

Questions 7 to 10 4 marks each

The coordinates of vertex P of this polygon of constraints are (16, 16).

The company must sell 300 chairs and 70 tables to maximize its revenue.

The minimum amount of time that Sandy needs to drive the guests to their hotel and return to the airport is 60 minutes.

Graph representing this situation

1 4

2 5

3 6

7

8

9

10

PART C: Extended Application Questions Questions 11 to 16 10 marks each (marked on 100% each according to rubric)

A COLOURFUL DISPLAY! EXAMPLE OF APPROPRIATE REASONING

GRAPHING SYSTEM OF INEQUALITIES AND POINTS

2x – 3y 6 3x +2y 12

x 0 3

y -2 0

Try point (0, 0):

60

600

6y3x2

FALSE!

Shade the sector that does not contain the

point (0, 0).

x 0 4

y 6 0

Try point (0, 0):

120

1200

12y2x3

FALSE!

Shade the sector that does not contain the

point (0, 0).

CONCLUSION

Only, Anna is right because the point indicated by her is situated in the region where the two semi-planes are super-imposed; all of the fireworks will be visible in this region.

11

AN INVESTMENT IN STOCKS

EXAMPLE OF APPROPRIATE REASONING

MINIMUM AMOUNT OF MONEY INVESTED TO BUY SHARES ON MONDAY

VERTEX AMOUNT OF MONEY: 40x + 8y

A(20,20) $960

B(10,60) $ 880 Minimum

C(80,80) $3 840

D(70,30) $3 040

NUMBER OF SHARES BOUGHT ON TUESDAY

Number of software company shares bought: 3 10 = 30

Number of mining company shares bought: 60 – 20 = 40

AMOUNT OF MONEY INVESTED TO BUY SHARES ON TUESDAY

$50 30 + $7 40 = $1780

AMOUNT OF MONEY INVESTED TO BUY SHARES ON MONDAY AND TUESDAY

$880 + $1780 = $2660

CONCLUSION

Joanne invested $2660 to buy shared for this client on Monday and Tuesday.

12

THE MUNICIPAL CAMPGROUND EXAMPLE OF APPROPRIATE REASONING

MAXIMUM POSSIBLE DAILY REVENUE FOR THE SUMMER OF 2009

VERTEX REVENUE: 25x + 35y

P(30,0) $750

Q(30,40) $2 150

R(50,50) $3 000 Maximum

S(100,0) $2 500

MODIFIED POLYGON OF CONSTRAINTS

Inequality representing the additional constraint: y ≤ 40

The coordinates of the new vertex are: (60, 40)

MAXIMUM POSSIBLE REVENUE FOR THE SUMMER OF 2010

VERTEX REVENUE: 25x + 35y

P(30,0) $750

Q(30,10) $2 150

(60,40) $2 900 Maximum

S(100,0) $2 500

DECREASE IN THE MAXIMUM POSSIBLE REVENUE

$3000 - $2900 = $100

CONCLUSION

The owner’s maximum possible daily revenue will decrease by $100 after agreeing to the residents’ demands.

13

A VISIT TO THE MUSEUM EXAMPLE OF APPROPRIATE REASONING

NEW CORRIDOR TO BE BUILT

In this graph, students must add an edge to obtain a circuit that travels over each edge only

once. This circuit exists if the degrees of all the vertices are even numbers.

VERTEX DEGREE

A 2

B 4

C 3 odd-numbered

D 2

E 3 odd-numbered

Students must an edge connecting vertices C and E to make the degree of each vertex an

even number. Therefore, the new corridor will be built between rooms C and E.

NEW GRAPH

MINIMIUM DISTANCE SOPHIE MUST TRAVEK DURING HER NEXT VISIT TO THE MUSEUM

CIRCUIT VALUE

A,B,C,E,A 69 Minimum

A,E,C,D,E,A 99

A,E,C E,A 76

A,B,C,E,A 71

CONCLUSION

During her visit to the museum, Sophie must travel a minimum distance of 69 m.

14

SNOW REMOVAL

EXAMPLE OF APPROPRIATE REASONING

TOTAL SNOW REMOVAL COSTS LAST YEAR Total length of roads needing snow removal: 12 + 10 + 12 + 15 + 5 + 11 + 8 + 8 = 81 km Total snow removal costs: 81 km $1000/km = $81 000

TOTAL LENGTHS OF ROADS NEEDING SNOW REMOVAL THIS YEAR We are looking for a tree of maximum value consisting of 5 edges connecting the 6 vertices of the graph representing the average number of users.

TOTAL SNOW REMOVAL COSTS THIS YEAR

Total length of roads needing snow removal: 11 + 15 + 8 + 5 + 12 = 51 km

Total snow removal costs: 51 km $1300/km = $66 300

DECREASE IN TOTAL SNOW REMOVAL COSTS $81 000 - $66 300 = $14 700 CONCLUSION

Compared with last year, total snow removal costs will decrease by $14 700 this year.

15

MULTITASKING AT THE AIRPORT

EXAMPLE OF APPROPRIATE REASONING

MINIMUM AMOUNT OF TIME NEEDED TO GET THE PLANE READY ON A NORMAL DAY

Luggage Inside plane Fuel

A. Unload luggage (15) B. Deplane passengers (10) D. Fill the fuel tank (20)

E. Load luggage (20) C. Clean the plane (15)

F. Board passengers (15)

G. Warm up the engine (5)

PATH VALUE

A,E,G 40

B, C, F, G 45 Minimum time (critical path)

D, G 25

MINIMUM AMOUNT OF TIME NEEDED TO GET THE PLANE READY TODAY

Luggage Inside plane Fuel

C. Unload luggage (15) D. Deplane passengers (10) E. Fill the fuel tank (20)

E. Load luggage (35) C. Clean the plane (15)

F. Board passengers (15)

G. Warm up the engine (5)

PATH VALUE

A,E,G 55 Minimum time (critical path)

B, C, F, G 45

D, G 25

INCREASE IN AMOUNT OF TIME TO GET PLANE READY

55 minutes – 45 minutes = 10 minutes CONCLUSION The crew took 10 minutes more to get the plane ready.

16

Appendix

Evalu

ati

on

Cri

teri

a

Descriptive Chart for Evaluating Competency Appendix A Uses Mathematical Reasoning

Observable Indicators of Student Behaviour

Level 5 Level 4 Level 3 Level 2 Level 1

Cr3

Proper application

of mathematical

reasoning suited to

the situation

Takes every aspect of the situation into account.

Uses efficient strategies in applying his/her mathematical reasoning.

Uses mathematical concepts and processes that enable him/her to meet the requirements of the situation efficiently.

Takes the main aspects of the situation into account.

Uses effective strategies in applying his/her mathematical reasoning.

Uses mathematical concepts and processes appropriate for the situation.

Takes some aspects of the situation into account.

Uses a few effective strategies for certain steps in applying his/her mathematical reasoning.

Uses some mathematical concepts and processes appropriate for the situation.

Takes few aspects of the situation into account.

Uses few appropriate strategies in applying his/her mathematical reasoning.

Uses very few mathematical concepts and processes appropriate for the situation.

Takes no aspect of the situation into account.

Uses inappropriate strategies in applying his/her mathematical reasoning.

Uses mathematical concepts and processes that are inappropriate for the situation.

Cr2

Correct use of

concepts and

processes

appropriate to the

situation

Applies the chosen mathematical concepts and processes appropriately.

Applies the chosen mathematical concepts and processes appropriately, but makes minor errors (e.g. miscalculations, inaccuracies, omissions).

Applies the chosen mathematical concepts and processes, but makes some conceptual or procedural errors.

Applies the chosen mathematical concepts and processes, but makes several conceptual or procedural errors.

Applies mathematical concepts and processes inappropriately, making many conceptual or procedural errors.

Cr4

Proper organization

of the steps in an

appropriate

procedure

Presents a complete and organized procedure that explicitly outlines what was done or how it was done.

Presents a complete and organized procedure that explicitly outlines what was done or how it was done, even though some of the steps are implicit.

Presents a procedure that is not very explicit about what was done or how it was done, because the work is unclear or not very organized.

Presents a procedure consisting of isolated elements, showing little or no work that explicitly outlines what was done or how it was done.

Presents a procedure that is completely unrelated to the situation or does not show any procedure.

Cr5

Correct justification

of the steps in an

appropriate

procedure

When required to justify or support his/her statements, conclusions or results, uses solid mathematical arguments.

Rigorously observes the rules and conventions of mathematical language.

When required to justify or support his/her statements, conclusions or results, uses appropriate mathematical arguments.

Observes the rules and conventions of mathematical language, despite some minor errors or omissions.

When required to justify or support his/her statements, conclusions or results, uses some appropriate mathematical arguments or uses rudimentary mathematical arguments.

Makes some errors or is sometimes inaccurate in using the rules and conventions of mathematical language.

When required to justify or support his/her statements, conclusions or results, uses only slightly appropriate mathematical arguments.

Makes several errors related to the rules and conventions of mathematical language.

When required to justify or support his/her statements, conclusions or results, uses erroneous or inappropriate mathematical arguments

Shows little or no concern for the rules and conventions of mathematical language.

Cr1

Formulation of a

conjecture

appropriate to the

situation

Formulates an astute conjecture based on a rigorous analysis of the situation or on examples that consider every aspect of a situation.

Formulates an appropriate conjecture based on a fitting analysis of the situation or on examples that consider most of the important aspects of the situation.

Formulates a partially appropriate conjecture based on an analysis of the situation or on examples that consider some aspects of the situation.

Formulates a conjecture that is not very appropriate, based on an analysis that considers few aspects of the situation, or on examples chosen purely by chance.

Formulates a conjecture that is unrelated to the situation.