TasThe 100 m Olympic Records
E43
Einstein High School and Newton High School compete in High School Olympic events in
different cities every two years.
Some students wondered if there was a relationship between the year of the event and the
winning times for the 100 m race. They also wanted to compare the winning times of the
two schools.
The following scatter plots represent the winning times for each school.
Time
(in seconds)
Einstein High School
(Times for the 100 m Race)
15.4
15.2
15.0
14.8
14.6
14.4
14.2
14.0
13.8
13.6
13.4
13.2
13.0
12.8
12.6
0
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000
Year
Time
(in seconds)
Newton High School
(Times for the 100 m Race)
15.4
15.2
15.0
14.8
14.6
14.4
14.2
14.0
13.8
13.6
13.4
13.2
13.0
12.8
12.6
0
1960 1964 1968 1972 1976 1980 1984 1988 1992 1996 2000
Year
Grade 9 Assessment of Mathematics
1
a) From the graph, determine the year Einstein High School ran 100 m with a
time of 13.4 s.
b) Draw a line of best fit on both scatter plots.
c) Describe how you decided where to draw your lines of best fit for the relationships.
d) i) Which line of best fit is steeper?
ii) Describe what this steeper slope tells you about how the Einstein High School
times are changing compared to the Newton High School times.
2
Grade 9 Assessment of Mathematics
e) There were no High School Olympics in 1982. By using the graph, Sajan predicted that
if there had been Olympics in 1982, the time for Newton High School would have been
approximately 14.3 s. Pat says that he thinks the time would have been closer to
14.0 s.
Write the prediction that is more reliable. Give reasons for your answer.
f) A local newspaper had a news story that claimed “Newton High School will never run
as fast as Einstein High School.”
Do your scatter plots support this claim? Give reasons for your answer.
Grade 9 Assessment of Mathematics
3
TasBabysitting Service
6254 P5Y
Danielle has a babysitting service.
The amount she charges consists of
• a flat fee of $10.00 for transportation
• plus $5.00 per hour.
a) Complete the table of values.
KU – N01
Number of hours, n
Amount charged, A
($)
1
15.00
2
3
4
5
35.00
6
4
Grade 9 Assessment of Mathematics
b) Create a scatter plot and join the points to show the relationship between the amount
charged, A, in dollars, and the number of hours, n.
KU – R12
Amount Charged vs. Number of Hours
A
60
55
50
Amount charged ($)
45
40
35
30
25
20
15
10
5
n
0
1
2
3
4
5
6
7
8
Number of hours
c) Write an equation for the relationship between the amount charged, A, and the
number of hours that Danielle babysits, n.
Grade 9 Assessment of Mathematics
AP – R15
5
d) Jolina also has a babysitting service. She charges $7.50 per hour. KU R12
Complete the amount charged column in the table of values below.
Number of hours, n
Amount charged, A
($)
1
7.50
Finite differences
2
3
4
30.00
5
e) Complete the finite differences column in the table of values above.
f) Is the relationship between the amount charged and the number of hours linear or
non-linear?
Circle one: linear or non-linear
Give reasons for your answer.
6
Grade 9 Assessment of Mathematics
KU-R21
g) The Smiths need a babysitter for New Year’s Eve.
PS – N31
They will leave at 4 p.m. and expect to return home at 3 a.m.
Jolina charges twice her hourly rate after midnight.
Should the Smiths hire Danielle or Jolina?
Show your work.
Hint:
Danielle charges
• a flat fee of $10.00 for
transportation
• plus $5.00 per hour.
Jolina charges
• $7.50 per hour
• and double after midnight.
Grade 9 Assessment of Mathematics
7
TasGumball Machine
905 M1N
The upper part of a gumball machine is a cylinder.
A company needs to estimate how often it must refill the machine.
a) Each gumball has a diameter of 1 cm.
Calculate the volume of one gumball.
Show your work. KUM14FPE
Zoom
one gumball
1 cm
Hint:
V=
8
!" πr3
Grade 9 Assessment of Mathematics
b) Sunreeta and Jodi are trying to figure out how many gumballs are in the machine.
Sunreeta claims that the following calculation will give the correct number of
gumballs.
Number of gumballs in the machine =
volume of the cylinder
volume of one gumball
Jodi disagrees, stating, “It’s not that simple.”
Which of the two girls do you agree with? Give reasons for your answer.
AP N8FCE/M20
c) A top view of the gumball machine is shown below. Estimate the diameter of the
machine by counting the gumballs.
KUN1/N2FEP
Top of Gumball Machine
Zoom
one gumball
1 cm
Grade 9 Assessment of Mathematics
d
9
d) The height of the cylinder, h, is the same as its diameter, d. Therefore, h = d.
Use your estimate for the diameter from question c) to calculate the volume of the
cylinder.
Show your work.
AP M20
d
Hint:
V = πr2h
e) Make your own estimate of the number of gumballs in the machine.
Describe how you got your estimate.
PS N1/N2FEP
f) The gumball machine must never be less than 20% full.
Approximately 50 gumballs are purchased each day from the machine.
Use your estimate from question e) to calculate how often the company must refill
the dispenser.
Show your work. PS N5C/N6P
10
Grade 9 Assessment of Mathematics
h
TasMatch Me
6709 P3X
Daniel walks in front of a motion sensor.
The graph below represents the relationship between
time, t, in seconds, and distance from the motion sensor,
D, in metres.
Distance vs. Time
9
D
Distance (m)
8
7
6
5
4
3
2
1
t
0
1
2
3
4
5
6
7
8
9
10
Time (s)
a) Describe Daniel’s walk. Include as many mathematical details as possible.
KU – R20
Hint:
Include information
about
• distance
• time
• speed
• direction
Grade 9 Assessment of Mathematics
11
b) Junaid walks in front of the motion sensor, trying to recreate Daniel’s graph.
His graph does not look exactly the same.
Distance vs. Time
9
D
k
7
’s
w
al
6
w
al
ni
k
el
5
na
id
4
’s
Da
Distance (m)
8
Ju
3
2
1
t
0
1
2
3
4
5
6
7
8
9
10
Time (s)
Use the table below to compare Junaid’s walk with Daniel’s walk.
Similarities
12
Differences
Grade 9 Assessment of Mathematics
AP – R22
c) Micheline tries to recreate Daniel’s walk. She starts 1 m away from the motion sensor
then walks 1 m away from it every 2 s.
i) Complete the table below.
AP-R12
Time
(s)
Distance from motion detector
(m)
(start)
0
1
2
4
6
8
ii) Graph Micheline’s walk below.
KU-R12
Distance vs. Time
9
D
8
k
’s
w
al
6
ni
el
5
Da
Distance (m)
7
4
3
2
1
t
0
1
2
3
4
5
6
7
8
9
10
Time (s)
d) Explain how Micheline should change her walk to make her graph look exactly like
Daniel’s graph.
PS – R22
Grade 9 Assessment of Mathematics
13
TasParallelograms
965 C6N
Three students each construct a different parallelogram using dynamic geometry software.
In each parallelogram, ∠A, ∠B, ∠C and ∠D are bisected and the lines are extended to
intersect the opposite sides.
The three results are shown below.
B
B
C
R
Q
S
Q
A
R
P
A
P
D
S
B
C
D
A
Q
R
P
S
C
D
a) Look at the shaded shape, PQRS, in each diagram above. Make a hypothesis about the
shape of PQRS.
14
PS M23C
Grade 9 Assessment of Mathematics
b) ABCD is a parallelogram where ∠A = 60°. Find the measures of ∠B, ∠C and ∠D and
write them on the diagram below. Show your work.
B
KU M22
C
60º
A
D
c) Four line segments are drawn to bisect each of the interior angles of the parallelogram
ABCD in question b).
Write the measures of a, b, c and d on the diagram below. Show your work.
B
c
c
bb
A
a
a
d
KU M22
C
d
D
Grade 9 Assessment of Mathematics
15
d) For each shaded triangle, use the angles you found in question c) to find the
measure of one of the angles in PQRS. In each case, give reasons for your answer.
AP M22
i)
B
C
∠PQR = ____________
R
Reasons:
S
Q
P
A
ii)
D
B
C
∠_____ = ____________
R
Reasons:
S
Q
P
A
iii)
D
B
C
∠_____ = ____________
R
Reasons:
S
Q
P
A
iv)
D
B
C
R
Q
Reasons:
S
P
A
16
∠_____ = ____________
D
Grade 9 Assessment of Mathematics
e) Look at your results from question d).
i) Do your results support your hypothesis from question a)?
Give reasons for your answer.
PS M25C
ii) A classmate says that your hypothesis works only for parallelograms in which one
of the angles is 60°.
Describe the additional work you would need to do in order to check whether your
hypothesis is true for any parallelogram. PS M25C
Grade 9 Assessment of Mathematics
17