Mathematics 506 SCI

Mid-Year Examination

January 2011

Administration & Marking Guide

565-506

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: Science option (SN5). 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

The skateboard ramp

Square root function

System of equations

Solving a second degree equation in one variable

Question 12

The circus comes to town

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 13

Wally world stock

Absolute value function

Solving absolute value equations in one variable

Interpreting and representing real-world situations using

different registers of representation

Question 14

Acid Concentration

Rational function

Solving rational equations in one variable

Interpreting and representing real-world situations using different registers of representation

Question 15

Path of a Cat’s Toy

Piecewise function : linear function, square root function, absolute value function and rational function

Solving linear, square root, absolute value and rational equations in one variable

Question 16

Operations on functions

Performs operations on functions

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 11th 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

B C

C A

A C

PART B: Short-Constructed Answer Questions

Questions 7 to 10

A) 6

B) -2 ( or -2 - 2

2 marks each correct answer 1 mark each correct answer not simplified 0 mark each incorrect answer

The rule for the inverse of the function is

4 marks for inverse and restrictions 3 marks inverse is found without restrictions 0 mark incorrect answer

dom f = ]-, 5[ ]5, [ or \{5}

ran f = ]-, 6[ ]6, [ or \{6}

2 marks each correct answer 0 mark each incorrect answer

A) The rule of correspondence is E(t) = 4|t 3| + 6.

B) The vertex of E(t) is (3, 6)

Accept any equivalent equation.

4 marks for correct rule and vertex 3 marks for correct rule 1 mark for correct/appropriate vertex 0 mark incorrect answers

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)

THE SKATEBOARD RAMP

EXAMPLE OF APPROPRIATE REASONING

EQUATION OF THE RAMP

where b = 1 and the vertex is (1,63)

Replacing point (10, 54), into the equation

DETERMINATION OF THE POINT OF ATTACHMENT OF THE BAR TO THE RAMP

Comparison method

Zeroes

Height of the bar

CONCLUSION

The height of the metal bar is 51 units from the ground.

11

THE CIRCUS COMES TO TOWN

EXAMPLE OF APPROPRIATE REASONING

MAXIMUM POSSIBLE PROFIT PER SHOW IN JUNE

VERTEX PROFIT: 15x + 25y

A (100,50) $2 750

B (100,200) $6 500

C (200,400) $13 000 Maximum profit per show in June: $13 000

D (550,50) $9 500

POLYGON OF CONSTRAINTS REPRESENTING THE NUMBER OF TICKETS THAT CAN BE SOLD FOR

EACHSHOW IN JULY AND AUGUST

MAXIMUM POSSIBLE PROFIT PER SHOW IN JULY AND AUGUST

VERTEX PROFIT: 15x + 25y

A (100,50) $2 750

E (100,500) $14 000 Maximum profit per show in July and August:

D (550,50) $9 500 $14 000

MAXIMUM TOTAL PROFIT THAT CAN BE EXPECTED IN THESE 3 MONTHS

June: $13 000 20 shows $260 000 Maximum total profit that can be

July: $14 000 20 shows $280 000 expected in these 3 months: $820 000

August: $14 000 20 shows $280 000

CONCLUSION

In these 3 months, circus management can expect to make a maximum total profit of $820 000.

12

The constraint imposed in June no longer applies. The polygon of constraints is no longer bounded by segment BC associated with the inequality y ≤ 2x. Segments AB and DC are extended to E. Polygon of constraints AED on the right represents the number of tickets that can be sold for each show in July and August.

COORDINATES OF POINT E

Equation of AB : x= 100 Equation of DC: x + y = 600 100 + y = 600 y = 500 Coordinates of point E : E (100,500)

WALLY WORLD STOCK

EXAMPLE OF APPROPRIATE REASONING

EQUATION OF THE ABSOLUTE VALUE FUNCTION

Use symmetry to find h

Points (8,13) and (12,13)

To find a, use points (0,7) and (8,13)

4

3

8

6

08

713

a

Since the function opens down, a is negative.

4

3a

Use any point to determine k:

kxxf 104

3)(

k 10124

313 Using point (12,13)

k 24

313

k2

313

5.14k

The equation is 5.14104

3)( xxf

DETERMINATION OF THE TIME WHEN STOCK IS WORTH $5.50

Solve using 5.5)( xf

5.14104

35.5 x

104

39 x

1012 x

Case 1 Case 2

22

1012

10

010

1

1

x

x

x

x

nRestrictio

2

1012

)10(12

10

010

2

2

2

x

x

x

x

x

nRestrictio

The only acceptable answer is 22.

CONCLUSION

The stock was worth $5.50 after 22 months.

13

ACID CONCENTRATION EXAMPLE OF APPROPRIATE REASONING

RULE OF FUNCTION f

Value of c

In this case, the rule of function f would be

Validating the rule using other data values in the table:

Rule of function f:

VOLUME OF PURE WATER TO BE ADDED TO OBTAIN A SOLUTION WITH AN ACID CONCENTRATION OF 5%

Find the value of x for which f x5.

CONCLUSION Nick must add 420 mL of pure water to the initial solution to obtain a solution with an acid concentration of 5%.

14

PATH OF A CAT’S TOY

EXAMPLE OF APPROPRIATE REASONING

DETERMINE RULE OF SQUARE ROOT FUNCTION Using linear function and x = 5

3)5(

13)5(2)5(

132)(

f

f

xxf

The rule of the square root function is 352)( xxf

DETERMINE RULE OF ABSOLUTE VALUE FUNCTION Using square root function and x = 14 to determine point of absolute value function

9)14(

35142)14(

352)(

f

f

xxf

Point (14, 9) belongs to absolute value function

Using point (14, 9) to determine parameter a1

3

8

38

117149

117)(

1

1

1

1

a

a

a

xaxf

The rule of the absolute value function is 1173

8)( xxf

15

DETERMINE RULE OF RATIONAL FUNCTION Using absolute value function and x = 23 to determine point of rational function

171161)6(3

8)23(

117233

8)23(

1173

8)(

f

f

xxf

Point (23, 17) belongs to rational function

Using point (23, 17) to determine parameter a2

36

312

52023

17

520

)(

2

2

2

2

a

a

a

x

axf

The rule of the rational value function is 520

36)(

xxf

DETERMINE DURATION OF TOY’S PATH

32

1220

36)20(3

20

363

520

368

520

36)(

x

x

x

x

x

xxf

CONCLUSION The duration of the path of the cat’s toy is 32 seconds.

OPERATIONS ON FUNCTIONS

EXAMPLE OF APPROPRIATE REASONING

DETERMINE RESULT OF (2x – 1) ((3x2 + 5) (4x)) (2x – 1) (12x3 + 20x) (24x4 + 40x2 – 12x3 – 20x) = 24x4 – 12x3 + 40x2 – 20 x

DETERMINE RESULT OF

((2x – 1) (3x2 + 5)) (4x)

(6x3 + 10x – 3 x2 – 5) (4x) (24x4 + 40x2 – 12x3 – 20x) = 24x4 – 12x3 + 40x2 – 20x

DETERMINE RESULT OF

((3x2 + 5) (4x)) (2x – 1) (12x3 + 20x) (2x – 1) 24x4 – 12x3 + 40x2 – 20x

CONJECTURE

The multiplication of functions is associative and commutative.

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.

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.