Mathematics 504 CST

Mid-Year Examination

January/February 2012

Administration & Marking Guide

563-504

Administration Guide

Design Team: EMSB

1. Presentation of the evaluation situation

This evaluation situation is consistent with principles regarding the evaluation of learning as outlined by the Ministry of Education, Leisure and Sport (MELS). Developed in conjunction with teachers as well as consultants from various school boards in Quebec, it is intended to give teachers some indication of the extent to which students have developed the competency Uses Mathematical Reasoning. This evaluation situation comprises three parts. Part A consists of six multiple choice questions and Part B consists of four short answer questions. These sections are intended to evaluate the student’s mastery of mathematical concepts and processes. Part C consists of six application tasks that focus on evaluating the competency Uses Mathematical Reasoning. The preliminary result for Part C is expressed as a mark out of 600, whereas the final result is expressed as a mark out of 60 which is calculated by dividing the preliminary result by 10 and rounding to the nearest unit. The student's total mark is the sum of results for Parts A, B, and C.

Type of Task Number of Questions Marks per Question Total Marks

Part A Multiple Choice 6 4 24

Part B Short Answer 4 4 16

Part C Application 6 10 60

Tasks in this evaluation situation focus on the main concepts and processes covered in year three of the Secondary Cycle Two Mathematics Program: Cultural, Social and Technical Option. This guide provides information about scoring student work on tasks that make up the evaluation situation. This guide also includes examples of appropriate solutions for each task in Part C. Appendix includes a rubric for the competency Uses Mathematical Reasoning (Appendix A).

1.1 Description of the documents

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) Question/Answer Booklet for the situations focusing on Competencies 2 (Uses Mathematical Reasoning).

1.2 Description of the tasks: connections to the Quebec Education Program (QEP)

Description of Part C For each task, the table below gives a brief description of the concepts and processes that a student may be required to mobilize in order to demonstrate the level of their competency Uses Mathematical Reasoning.

Title of the situation Concepts and processes

Question 11

A Fundraiser for a Food Bank

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 12

A Fundraiser for the Prom

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

A Summer Job

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 14

A Design for a Tile

Observing geometric transformations in the Cartesian coordinate system

- Graphic representation

Constructs, in the Cartesian plane, the image of a figure using a transformation rule

Anticipates the effect of a geometric transformation

Defines algebraically the rule for a geometric transformation

Question 15

The Farmer’s Plots of Land

Finds the following unknown measurements, using properties of figures and relations: area of equivalent figures, segments resulting from equivalent figures

Question 16

The Candy Coating

Finds the following unknown measurements, using

properties of figures and relations: volume of

equivalent solids, segments resulting from equivalent

figures (solids)

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 February 3rd 2012.

3. Possible adaptations1

Adaptations are permitted for this evaluation situation, but none of them involve changing its content. In fact, any change in the content of the evaluation situation, such as removing or changing a requirement, would compromise its validity.2 Adaptations are intended for students with learning difficulties or social maladjustments, or for students who have temporary limitations due to illness or special circumstances. The school could make these adaptations for students who require special measures. It should be noted that adaptations must always be made in order to allow a student to demonstrate their level of competency development, but must in no way compromise the validity of the evaluation situation. In other words, the adaptations should consist of measures related to the administration of the evaluation situation, its format, or the way in which students submit their work.

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 in part C.

Remind students that in Part C the scorer must give a mark of 0 to students who fail to show their work or whose work does not justify their answer.

4.2 General procedure

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

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.

5. Administration of the evaluation situation:

• Hand out copies of the Question/Answer Booklet. Ask students to go through the

document, familiarizing themselves with all of the information and requirements. Make sure they know where they must write their names answers, calculations, and explanations.

• Ask students to read the evaluation criteria used to evaluate their level of competency

development. • Describe the basic rules for evaluation situations:

o Each student works alone. o Students may use a calculator but are expected to indicate the sequence of

operations involved as part of the justification for their solution. o Students have three hours to complete the evaluation situation. o Students may use resources such as a geometry set, graph paper and the

handwritten memory aid. o During the evaluation situation, the teacher may clarify the meaning of general

vocabulary related to the context of the task. • When time is up, collect the examination booklets.

6. Using the results

The procedure outlined below should be used to help teachers evaluate the student's mathematical competency Uses Mathematical Reasoning by taking into account the information collected during the administration of the evaluation situation. Examples of appropriate reasoning are given for each task. The student’s reasoning may be different, yet still meet the requirements of the task. The scorer must exercise judgment and accept other appropriate reasoning. A judgment regarding the student’s work on each task is made by taking into account the student’s performance level for each evaluation criterion considered. The level achieved for criterion 3 is the maximum possible level for criteria 2, 4 and 5. The scorer must give a mark of zero to students who fail to show their work or whose work does not justify their answer.

The preliminary result for Part C is expressed as a mark out of 600, whereas the final result is expressed as a mark out of 60 which is calculated by dividing the preliminary result by 10 and rounding to the nearest unit. The student's total mark is the sum of results for Parts A, B, and C. Referring to this evaluation situation, make a judgment regarding the student's level of competency development for the competency Uses Mathematical Reasoning.

7. Marking Key

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

1. D

2. A 3. C 4. A 5. B 6. C

PART B: Short-Constructed Answer Questions

Questions 7 to 10

7. The system of inequalities is: Let x be the number oranges Let y be the number of grapefruits

2 marks per inequality

8. Polygon of constraints:

There are 12 solutions.

4 marks; correct answer 1 mark; 4 solutions

0 mark; incorrect answer

9. The rule of the transformation that associates the two curves is (x,y) (x, -6y).

4 marks; correct answer 0 mark; incorrect answer

10. The perimeter of the rectangle measures 68 cm.

4 marks; correct answer 1 mark; found the length as 25 cm

0 mark; incorrect answer

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

11. A FUND RAISER FOR A FOOD BANK

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF SYSTEM OF INEQUALITIES

Let x : the number of cakes sold and y : the number of pies sold

x 0 y 0

x 10 y 45

x + y 80 y 3x

POLYGON OF CONSTRAINTS AND ITS VERTICES

DETERMINATION OF FUNCTION RULE AND MAXIMUM REVENUE

VERTICES OF POLYGON

OF CONSTRAINTS

R = 12 x + 9 y 300

A (10, 45) $225

B (10, 70) $450

C (20, 60) $480 Maximum

D (15, 45) $285

CONCLUSION

The maximum amount that the students can raise for the food bank is $480.

Num

ber

of

pie

s so

ld

Number of cakes sold

POSSIBLE COMBINATIONS OF SALES

VERTICES OF POLYGON OF CONSTRAINTS

A (10, 45)

B (10, 70)

C (20, 60)

D (15, 45)

12. A FUND RAISER FOR THE PROM

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF VERTICES OF POLYGON OF CONSTRAINTS

A (20, 25) B (20, 50) C (40, 50) D (60, 30) E (30, 15)

DETERMINATION OF FUNCTION RULE AND MAXIMUM PROFIT

Function Rule : P = 3 x + 5 y

VERTICES OF POLYGON

OF CONSTRAINTS

P = 3 x + 5 y

A (20, 25) $185

B (20, 50) $310

C (40, 50) $370 Maximum

D (60, 30) $330

E (30, 15) $165

DETERMINATION OF NUMBER OF PAIRS OF GLOVES AND SCARVES SOLD BY THE

THREE FRIENDS

Benjamin made the maximum profit; he sold 40 pairs of gloves and 50 scarves.

Number of pairs of gloves sold by Moritz: 40 10 = 30

Number of scarves sold by Moritz: 50 2 = 25

Number of pairs of gloves sold by Melinda: 2(40) – 25 = 55

Number of scarves sold by Melinda: 50 – 15 = 35

DETERMINATION OF MONEY RAISED BY THE THREE FRIENDS Benjamin made the maximum profit: $370. Money raised by Moritz: P = 3(30) + 5(25) = 90 + 125 = $215 Money raised by Melinda: P = 3(55) + 5(35) = 165 + 175 = $340 Total amount of money raised: $370 + $215 + $340 = $925

CONCLUSION The total amount of money the three friends raised for their prom is $925.

13. A SUMMER JOB

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF FUNCTION RULE AND MAXIMUM REVENUE IN MAY

R = 7.50 x + 9 y

POLYGON OF CONSTRAINTS OF POSSIB LE WORK HOURS IN JUNE AND JULY

New constraint : x + y ≤ 70. There are two new vertices in the polygon of constraints. Polygon of constraints ABJKD represents the possible work hours in June and July.

MAXIMUM REVENUE IN JUNE AND JULY AND TOTAL REVENUE IN THE 3 MONTHS

Total Revenue in the 3 Months

$435 + 2 x $585 = $1 605

CONCLUSION Prasanth will have earned $1 605 in the three months prior to his trip.

VERTEX REVENUE : 7.50 x + 9 y

A (10, 20) $254

B (10, 40) $435 Maximum

C (25, 25) $412.50

D (15, 15) $247.50

VERTEX REVENUE : 7.50 x + 9 y

A (10, 20) $254

B (10, 40) $435

J (30, 40) $585 Maximum

D (35, 35) $577.50

D (15, 15) $247.50

Coordinates of Point J: (30, 40)

Coordinates of Point K: (35, 35)

14. A DESIGN FOR A TILE

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF COORDINATES OF INITIAL FIGURE

VERTEX COORDINATES

A (-6, 4)

B (0, - 6)

C (0, -2)

D (4, 2)

DETERMINATION OF COORDINATES OF IMAGE 1 UNDER THE TRANSFORMATION

VERTEX COORDINATES

A’ (-3, 2)

B’ (0, - 3)

C’ (0, -1)

D’ (2, 1)

A

B

C

D A'

B'

C'

D'

x

y

DETERMINATION OF TRANSLATION RULE TO PRODUCE IMAGE 2 We want vertex B’ to move so that it is located at vertex A. Coordinates of vertex B’: (0, - 3) to be translated to location (-6, 4); therefore, it must be translated 6 units left (-6) and 7 units up (+7) The transformation rule is :

DETERMINATION OF COORDINATES OF IMAGE 2

VERTEX COORDINATES

A’’ (-9, 9)

B’’ (-6, 4)

C’’ (-6, 6)

D’’ (-4, 8)

DETERMINATION OF TRANSLATION RULE TO PRODUCE IMAGE 2 We want vertex B’’ to move so that it is located at vertex D. Coordinates of vertex B’’: (-6, 4) to be translated to location (4, 2); therefore, it must be translated 10 units right (+10) and 2 units down (-2) The transformation rule is :

DETERMINATION OF COORDINATES OF IMAGE 3

VERTEX COORDINATES

A’’’ (1, 7)

B’’’ (4, 2)

C’’’ (4, 4)

D’’’ (6, 6)

FINAL DESIGN OF THE TILE

x

y

A

B’’

D’’

C’’

A’’

A’’’

C’’’

D’’’

B’’’ D

C

CONCLUSION The transformation rule to move image 1 to produce image 2 is . The transformation rule to move image 2 to produce image 3 is

B

15. THE FARMER’S PLOTS OF LAND

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF AREA OF RIGHT PLOT OF LAND

Missing measures for small rectangle: top length: 48 m – 32 m = 16 m top left width: 36 m – 26 m = 10 m

Area of small rectangle = 16 m x 10 m = 160 m2 Area of larger rectangle = 48 m x 26 m = 1 248 m2 Total area of right plot of land = 1 248 m2 + 160 m2 = 1 408 m2

DETERMINATION OF DIMENSIONS OF LEFT PLOT OF LAND Figure can be divided into a rectangle and a trapezoid. Area of rectangle : (6x) (2x) = 12x2 Dimensions of trapezoid: long base : 6x; small bases: 4x, height: 2x Area of trapezoid: (6x + 4x) (2x) = 20 x2 = 10 x2 2 2 Total area of left plot: 12 x2 + 10 x2 = 22 x2

Because both plots of land are equivalent: area of left plot = area of right plot 22 x2 = 1408 m2 x2 = 64 m2 So x = 8 m Dimensions are 48 m, 32 m, 32 m, 20.8 m and 16 m.

DETERMINATION OF COST TO FENCE BOTH PLOTS OF LAND

Perimeter of left plot of land: 48 m + 32 m + 32 m + 20.8 m + 16 m = 148.8 m Perimeter of right plot of land: 48 m + 26 m + 32 m + 10 m + 16 m + 36 m = 168 m Total length of fencing required: 148.8 m + 168 m = 316.8 m

Cost of fencing required = 316.8 m $40/m = $12 672 CONCLUSION

It will cost $12 672 to fence both plots of land.

16. THE CANDY COATING

EXAMPLE OF AN APPROPRIATE SOLUTION

DETERMINATION OF VOLUME OF SPHERE Knowing the surface area of the sphere, we can determine the radius of the sphere Surface area of sphere: 4πr2= 201.06 cm2

Volume of sphere: 268.08 cm3

DETERMINATION OF RADIUS OF CYLINDER A Because cylinder A is equivalent to the sphere, its volume is 268.08 cm3.

Volume of sphere = area of base x height= = 268.08 cm3

3.77 cm

DETERMINATION OF DIMENSIONS OF CYLINDER B

Using the similarity ratio of volumes 343/8, the similarity ratio of sides is

= 7/3

The height of cylinder B is

x = 2.57cm

The radius of cylinder B is

x = 1.62 cm

DETERMINATION OF SURFACE AREA OF CYLINDER B

Surface area of cylinder = Area of the bases + lateral area = 2

=2 ) = 2 8.24 + 26.16 = 16.48 + 26.16 = 42.64 cm2

DETERMINATION OF COST OF COATING 500 PIECES OF CYLINDER B Surface area of 500 pieces: 500 x 42.64 cm2 = 21 320 cm2 Cost of coating = $2.50/1 000 cm2 x 21 320 cm2 = $53.30

CONCLUSION It will cost the confectioner $53.30 to cover 500 pieces of cylinder B chocolates with candy coating.

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.

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.

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.