Secondary Cycle Two Year One June 2008 Competency 2 and Competency 3 Situations

Transcription

1 Mathematics Examination Secondary Cycle Two Year One June 2008 Competency 2 and Competency 3 Situations Time: 3 hours Student Booklet Name : Group : June 2008

2 The following criteria will be used to evaluate your level of competency development in the different situations presented in this booklet. Competency 2: Evaluation Criteria Uses Mathematical Reasoning Cr1 - Formulation of a conjecture appropriate to the situation Cr2 - Correct application of the concepts and processes appropriate to the situation Cr3 - Proper implementation of mathematical reasoning suited to the situation Cr4 - Proper organization of the steps in a proof suited to the situation Cr5 - Correct justification of the steps in a proof suited to the situation Competency 3: Evaluation Criteria Communicates By Using Mathematical Language Cr1 - Correct translation of a mathematical concept or process into another register of semiotic representation Cr2 Correct interpretation of a mathematical message involving at least two registers of semiotic representation Cr3 Production of a message appropriate to the communication context Cr4 Production of a message in keeping with the terminology, rules and conventions of mathematics Secondary Cycle Two Year One Page 1

3 Instructions 1. Provide all the required information in the spaces in this booklet. 2. There are 12 questions in this booklet. For each question, you must demonstrate your reasoning to justify your answer. The steps in your procedure must be organized and clearly presented. 3. You are permitted to use graph paper, a ruler, a compass, a set square, a protractor and a calculator. 4. You may refer to the memory aid you prepared on your own before the examination. The memory aid consists of one letter-sized sheet of paper (8.5 11). Both sides of the sheet may be used. Any mechanical reproduction of this memory aid is forbidden. All other reference materials are forbidden. Note: Figures are not necessarily drawn to scale. Secondary Cycle Two Year One Page 2

4 1. ICE CREAM DILEMMA At one of the stops during the Amazing Race, organizers plan to serve ice cream to both staff and contestants. There are 15 staff members and 28 contestants. The organizers have purchased five 1-L containers of ice cream and two boxes of ice cream cones. Each box has 24 cones. Every cone will have one scoop of ice cream. Each scoop is in the shape of a sphere with a diameter that is the same measure as the diameter of the top of the cone. Each cone has the following dimensions:! height 7.8 cm! apothem 8.4 cm 7.8 cm 8.4 cm Do they have enough ice cream? Secondary Cycle Two Year One Page 3

5 Show or explain how you found your answer. Do they have enough ice cream? YES " NO " Evaluation Criteria C2: Uses mathematical reasoning Observable indicators corresponding to level 1-5 Overall Cr3 Cr1 Cr2 Cr4 Cr5 Secondary Cycle Two Year One Page 4

6 2. A FISHY STORY Jeff s aquarium, in the shape of a rectangular prism, is filled to 80% of its height. He wants to add three solid food cones to the tank. He claims that the tank will not overflow as a result. The interior dimensions of the tank are:! length 40 cm! width 30 cm! height 25 cm The food cone has a radius of 9 cm and a height of 24 cm. 25 cm 30 cm 24 cm 40 cm 9 cm Is Jeff right? Secondary Cycle Two Year One Page 5

7 Show or explain how you found your answer. Is Jeff right? YES " NO " C2: Uses mathematical reasoning Observable indicators corresponding to level 1-5 Overall Cr3 Evaluation Criteria Cr1 Cr2 Cr4 Cr5 Secondary Cycle Two Year One Page 6

8 3. THE INVESTMENT BANKER Frank, John, and Jessica compare their respective savings. # Frank has $1000 and saves $5.00 per week. # John has $850 and saves $10.00 per week. # Jessica has $600 and saves $12.00 per week. After a certain period, Frank and John have the same amount in savings. Ten weeks after that period of time, how much does Jessica have? Secondary Cycle Two Year One Page 7

9 Show or explain how you found your answer. C2: Uses mathematical reasoning Observable indicators corresponding to level 1-5 Overall Cr3 Ten weeks after, Jessica has $ Evaluation Criteria Cr1 Cr2 Cr4 Cr5 Secondary Cycle Two Year One Page 8

10 4. THE INTERIOR DECORATORS Helen has been hired to paint a room in a building. The room has two rectangular walls and a curved wall. The ceiling is in the shape of a quarter of a circle. The top view of the room is given below. She has been asked to paint the ceiling, the curved wall and one of the rectangular walls. 8.5 m 8.5 m Each rectangular wall is 8.5 m long and 4 m high. A 3.8-litre can of paint will cover 40 m 2 with one coat. Two coats of paint are needed. How many cans of paint should Helen buy? Secondary Cycle Two Year One Page 9

11 Show or explain how you found your answer. C2: Uses mathematical reasoning Observable indicators corresponding to level 1-5 Overall Cr3 Helen should buy cans of paint. Evaluation Criteria Cr1 Cr2 Cr4 Cr5 Secondary Cycle Two Year One Page 10

12 5. TO THE POINT During the Amazing Race contest, participants were required to perform the following task. They had to throw darts at one of two targets. (A dart had to hit either the target or the backboard to count as a throw.) The targets are illustrated below. Most of the contestants believed that the circular target gave the highest probability of success. It turns out that they were correct. The two dartboards are illustrated below. 40 cm 50 cm 50 cm One board has a small square inside a larger square while the other has a circle inscribed in a square. Prove that the contestants are correct. Secondary Cycle Two Year One Page 11

13 Show or explain how you found your answer. C2: Uses mathematical reasoning Observable indicators corresponding to level 1-5 Overall Cr3 Evaluation Criteria Cr1 Cr2 Cr4 Cr5 Secondary Cycle Two Year One Page 12

14 6. GEOMETRIC GARDENS One of the stops for the participants of the Amazing Race is in Washington, D.C. They visit two beautiful flower gardens, one in the shape of a triangle and the other in the shape of a rectangle. The dimensions can be represented by algebraic expressions, as shown in the diagram below. 4x 8 2x 2 6x 14 6x 18 The brochure that John is reading about the gardens says that these gardens are equivalent in area. (In the U.S., the unit of measure that is used is the foot.) What are the dimensions of each garden? Secondary Cycle Two Year One Page 13

15 Show or explain how you found your answer. C2: Uses mathematical reasoning Observable indicators corresponding to level 1-5 Overall Cr3 Evaluation Criteria Cr1 Cr2 Cr4 Cr5 The triangular garden has a height of ft and a base of ft. The rectangular garden has a length of ft and a width of ft. Secondary Cycle Two Year One Page 14

16 Jose is thinking of a number. 7. JOSE S NUMBER # When he doubles the number, then subtracts 1, he has the width of a rectangle. # When he multiplies the original number by 4, then subtracts 3, he has its length. He asks his friend Chelsea what the length of the diagonal of that rectangle would be. Chelsea claims she does not have enough information so he gives her a hint: the perimeter of the rectangle is 46 units. What is the length of the diagonal of the rectangle? Secondary Cycle Two Year One Page 15

17 Show or explain how you found your answer. Evaluation Criteria C2: Uses mathematical reasoning Observable indicators corresponding to level 1-5 Overall Cr3 Cr1 Cr2 Cr4 Cr5 The length of the diagonal is units. Secondary Cycle Two Year One Page 16

18 8. JACOB S REPORT Jacob and his parents are upset because they think that the average on his report card is too low. When they calculate the average of Jacob s marks, they get 81.3%. Jacob s Report Card Subject Mark Credits Mathematics 84 6 French 76 6 English 78 6 Geography 80 4 History 75 4 Science 86 6 Phys. Ed Average Write a memo to Jacob s parents explaining how the computer arrived at 80.7% and why the school reports a weighted average rather than an arithmetic average. Secondary Cycle Two Year One Page 17

19 Your memo: Evaluation Criteria C3: Communicates by using mathematical language Observable indicators corresponding to level 1-5 Overall Secondary Cycle Two Year One Cr1 Cr2 Cr3 Cr4 Page 18

20 9. BUS VS MINIVAN A high school class is arranging a field trip. The organizers have narrowed their transportation options to the following two:! Option 1 Bus rental at $900/day! Option 2 Minivan rental at $150/day # There can be a maximum of 50 students on the trip. # They need at least 25 students to sign up # The bus could hold all the students and their chaperones # Each minivan can transport up to 8 students # The amount the students are charged must cover all the transportation costs (of students and chaperones) $150/day $900/day The organizers would like to analyze the cost of this trip per student, based on the transportation they choose and the number of students that sign up and pay. Create a table or tables of value detailing the relevant information and then write sentences that highlight at least 2 observations you can make from your table(s). Secondary Cycle Two Year One Page 19

21 Your table(s) of values and sentences: Evaluation Criteria C3: Communicates by using mathematical language Observable indicators corresponding to level 1-5 Overall Secondary Cycle Two Year One Cr1 Cr2 Cr3 Cr4 Page 20

22 10. Glitter Girl Sandra's little sister is making a poster. Her idea is to cut out a rectangle, cover it with glitter, and then attach other decorations to it. She wants the poster to be similar to a rectangle with a width of 4 cm and a length of 5 cm. The tube of glitter she is using can cover 720 cm2. She plans to use all of the glitter. What will be the dimensions of the poster she makes? Secondary Cycle Two Year One Page 21

23 Show or explain how you found your answer. Evaluation Criteria C2: Uses mathematical reasoning Observable indicators corresponding to level 1-5 Overall Cr3 Cr1 Cr2 Cr4 Cr5 The dimensions of the poster will be cm by cm. Secondary Cycle Two Year One Page 22

24 11. RICKSHAW RATES While in Japan, the Amazing Race contestants have to take a 15-km rickshaw ride. A rickshaw is a carriage pulled by a person. This means of transportation originated in Japan and is still used by tourists, much like the caleches in Montreal. Each group of contestants is given an envelope with information on four rickshaw companies. In Japan, the monetary units are Japanese Yen ( ). One Canadian dollar is approximately 100 Japanese Yen. Imperial Rickshaw Citizen Rickshaw Cost ( ) km Distance (km) Rickshaw ToGo C = d Where d = distance in km C = total cost ( ) Samuraï Rickshaw Charges 1500 flat rate for any ride over 10 km but less than 20 km. Which is the least expensive rickshaw company for the 15 km-ride? Secondary Cycle Two Year One Page 23

25 Show or explain how you found your answer. is the least expensive rickshaw company for the 15 km ride. Secondary Cycle Two Year One Evaluation Criteria C3: Communicates by using mathematical language Observable indicators corresponding to level 1-5 Overall Cr1 Cr2 Cr3 Cr4 Page 24

26 12. WHO S THE BEST? The Ironman Triathlon features a 3.9-km swim, a 180-km bike ride, and a complete marathon (42.2 km) all in succession. Athletes have 17 hours to complete the event. Below is a summary of the results of the 2446 competitors who finished the race in Penticton BC in These competitors came from all over the world to compete and 203 of them came from Quebec and Ontario. Use your understanding of statistics to comment on the performance of the competitors from Quebec and Ontario compared to the group as a whole. Use measures of central tendency as well as a graph, comparing completion times to the percentage of finishers from each population, to support your comments. Grouped Data Table: Completion times for Ironman competitors Hours to complete (Class) Number of finishers Percent of finishers QC & ON finishers [8,9[ 10 2 [9,10[ 54 8 [10,11[ [11,12[ [12,13[ [13,14[ [14,15[ [15,16[ [16,17[ Percent of QC & ON finishers Total finishers Mean 12.9 Median Class [12,13[ Modal Class [12,13[ Secondary Cycle Two Year One Page 25

27 Draw a graph and support your conclusions. Based on my understanding of statistics, the QC and Ontario finishers Evaluation Criteria C3: Communicates by using mathematical language Observable indicators corresponding to level 1-5 Overall Secondary Cycle Two Year One Cr1 Cr2 Cr3 Cr4 Page 26