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Mathematics Examination — 563-306
Secondary Cycle Two Year One June 2008
Competency 2 and Competency 3 Situations
Student Booklet
Name :
Group :
June 2008
Time: 3 hours
Secondary Cycle Two Year One – 563-306 Page 1
The following criteria will be used to evaluate your level of competency
development in the different situations presented in this booklet.
Evaluation Criteria Competency 2: 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
Evaluation Criteria Competency 3: 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 – 563-306 Page 2
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 – 563-306 Page 3
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
cm8.4 cm
Do they have enough ice cream?
Secondary Cycle Two Year One – 563-306 Page 4
Show or explain how you found your answer. Do they have enough ice cream?
YES " NO "
C2: Uses mathematical reasoning
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia
Cr3
Cr1
Cr2
Cr4
Cr5
Secondary Cycle Two Year One – 563-306 Page 5
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.
24 cm
9 cm
40 cm
30 cm
25 cm
Is Jeff right?
Secondary Cycle Two Year One – 563-306 Page 6
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
Eval
uatio
n C
riter
ia
Cr3
Cr1
Cr2
Cr4
Cr5
Secondary Cycle Two Year One – 563-306 Page 7
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 – 563-306 Page 8
Show or explain how you found your answer. Ten weeks after, Jessica has $______________
C2: Uses mathematical reasoning
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia
Cr3
Cr1
Cr2
Cr4
Cr5
Secondary Cycle Two Year One – 563-306 Page 9
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 m2 with one coat. Two coats of
paint are needed.
How many cans of paint should Helen buy ?
Secondary Cycle Two Year One – 563-306 Page 10
Show or explain how you found your answer. Helen should buy ______________ cans of paint.
C2: Uses mathematical reasoning
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia
Cr3
Cr1
Cr2
Cr4
Cr5
Secondary Cycle Two Year One – 563-306 Page 11
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 – 563-306 Page 12
Show or explain how you found your answer.
C2: Uses mathematical reasoning
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia
Cr3
Cr1
Cr2
Cr4
Cr5
Secondary Cycle Two Year One – 563-306 Page 13
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.
6x − 14 6x − 18
4x − 8 2x − 2
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 – 563-306 Page 14
Show or explain how you found your answer. 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.
C2: Uses mathematical reasoning
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia
Cr3
Cr1
Cr2
Cr4
Cr5
Secondary Cycle Two Year One – 563-306 Page 15
7. JOSE’S NUMBER
Jose is thinking of a 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 – 563-306 Page 16
Show or explain how you found your answer. The length of the diagonal is_______________ units.
C2: Uses mathematical reasoning
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia
Cr3
Cr1
Cr2
Cr4
Cr5
Secondary Cycle Two Year One – 563-306 Page 17
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. 90 2 Average 80.7 34
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 – 563-306 Page 18
Your memo:
C3: Communicates by using mathematical language
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia Cr1
Cr2
Cr3
Cr4
Secondary Cycle Two Year One – 563-306 Page 19
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 – 563-306 Page 20
Your table(s) of values and sentences:
C3: Communicates by using mathematical language
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia Cr1
Cr2
Cr3
Cr4
Secondary Cycle Two Year One – 563-306 Page 21
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 – 563-306 Page 22
Show or explain how you found your answer. The dimensions of the poster will be _______cm by ________cm.
C2: Uses mathematical reasoning
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia
Cr3
Cr1
Cr2
Cr4
Cr5
Secondary Cycle Two Year One – 563-306 Page 23
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.
300
0 100
200
400
500
600
700
Distance (km) 0 2 4 6 8 10 12
Cos
t (¥)
km ¥ 0 350 4 410 8 470 ... ...
Imperial Rickshaw Citizen Rickshaw
Rickshaw ToGo Samuraï Rickshaw
Which is the least expensive rickshaw company
for the 15 km-ride?
Charges ¥1500 flat rate for any ride over 10 km but less than 20 km.
C = 300 + 18d Where
d = distance in km C = total cost (¥)
Secondary Cycle Two Year One – 563-306 Page 24
Show or explain how you found your answer.
______________________is the least expensive rickshaw company for the 15 km ride.
C3: Communicates by using mathematical language
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia Cr1
Cr2
Cr3
Cr4
Secondary Cycle Two Year One – 563-306 Page 25
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 2007. 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
Percent of QC & ON finishers
[8,9[ 10 2
[9,10[ 54 8
[10,11[ 307 34
[11,12[ 470 37
[12,13[ 509 41
[13,14[ 406 35
[14,15[ 339 24
[15,16[ 227 14
[16,17[ 124 8
Total finishers 2446 203
Mean 12.9
Median Class [12,13[
Modal Class [12,13[
Secondary Cycle Two Year One – 563-306 Page 26
Draw a graph and support your conclusions.
Based on my understanding of statistics, the QC and Ontario finishers …
C3: Communicates by using mathematical language
Observable indicators corresponding to level… 1-5 Overall
Eval
uatio
n C
riter
ia Cr1
Cr2
Cr3
Cr4