Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91032
Exemplar for Internal Achievement Standard
Mathematics and Statistics Level 1
This exemplar supports assessment against:
Achievement Standard 91032
Apply right-angled triangles in solving measurement problems
An annotated exemplar is an extract of student evidence, with a commentary, to explain key
aspects of the standard. These will assist teachers to make assessment judgements at the
grade boundaries.
New Zealand Qualification Authority
To support internal assessment from 2014
© NZQA 2014
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91032
Grade Boundary: Low Excellence
1.
For Excellence, the student needs to apply right-angled triangles, using extended
abstract thinking, in solving measurement problems.
This involves one or more of: devising a strategy to investigate or solve a problem,
identifying relevant concepts in context, developing a chain of logical reasoning or
proof, forming a generalisation and also using correct mathematical statements or
communicating mathematical insight.
This student’s evidence is a response to the TKI assessment resource ‘What’s the
Angle?’
This student has devised a strategy to determine, the dimensions of the sail (1) and the
minimum height of the van (2). Correct mathematical statements have been used
throughout the response.
For a more secure Excellence, the student would need to correctly calculate the
minimum length of the van.
© NZQA 2014
Student 1: Low Excellence
Anchor point A (AB)
tan 20 =
AB
9
9tan20 = AB
AB = 3.28m
Length of sail edge (AC)
a2 + b2 = c2
AC2 = 3.282 + 92
AC2 = 10.76 + 81
AC = 91.76
AC = 9.58m (2dp)
Length of sail edge (AD)
The angle at which to cut the sail fabric at the corner (CDA)
AD2 = 9.582 + 32
tan D =
9.58
3
9.58
D = tan −1
3
AD2 = 91.78 + 9
D = 72.60
AD = 100.78
AD = 10.04m (2dp)
The sail is a right angled triangle with sides 3m, 9.58m and 10.04m. The angles are 90o, 72.60 and
17.40.
The length of the pole will be 3.28 + 2.5 = 5.78m
The minimum width of the van is 2.4m
sin 13 =
DC
5.78
DC = 5.78sin 13
DC =1.30m
Height = 130cm
2
The minimum height of the van is 130cm
AC
5.78
AC = 5.78 × cos13 = 5.63m
cos13 =
BC 2 = 5.632 + 2.4 2
BC = 37.4569
BC = 6.12m
The minimum length of the van is 6.12m
1
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91032
Grade Boundary: High Merit
2.
For Merit, the student needs to apply right-angled triangles, using relational thinking, in
solving measurement problems.
This involves one or more of: selecting and carrying out a logical sequence of steps,
connecting different concepts and representations, demonstrating understanding of
concepts, forming and using a model and also relating findings to a context, or
communicating thinking using appropriate mathematical statements.
This student’s evidence is a response to the TKI assessment resource ‘What’s the
Angle?’
This student has selected and carried out a logical sequence of steps to correctly
determine most of the dimensions of the sail (1) and the minimum dimensions of the
van without including the extra 2.5 m in the length of the pole (2). Appropriate
mathematical statements have been used throughout the response.
To be awarded Excellence, the student would need to determine all dimensions of the
sail, consider the extra 2.5 m in the length of the pole and identify the minimum
dimensions of the van.
© NZQA 2014
Student 2: High Merit
1.
tan 23 =
AB
8
AB = tan 23 x 8
AB = 3.4m (1dp)
2. AC2 = CB2 + AB2
AC2 = 82 + 3.42
AC = 64 + 11.56
AC = 8.7m (1dp)
3.
8.7
2
 8.7 
E = tan −1 

 2 
tan E =
E = 77.10 (1dp)
The Sail
sin 13 =
1
x
AB
x = sin13 x 3.4 = 0.76m
AD
AB
AD
cos13 =
3.4
cos13 =
2
AD = cos13 x 3.4 = 3.31m (2dp)
y2 = AD2 – AC2
y = (10.96 − 5.76)
y = 2.28m
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91032
Grade Boundary: Low Merit
3.
For Merit, the student needs to apply right-angled triangles, using relational thinking, in
solving measurement problems.
This involves one or more of: selecting and carrying out a logical sequence of steps,
connecting different concepts and representations, demonstrating understanding of
concepts, forming and using a model and also relating findings to a context, or
communicating thinking using appropriate mathematical statements.
This student’s evidence is a response to the TKI assessment resource ‘What’s the
Angle?’
This student has selected and carried out a logical sequence of steps in calculating the
lengths of AB (1) and AC (2) correctly, but there is a transfer error in calculating the
angle in the sail (3). The student has demonstrated relational thinking by considering
the correct length of the pole in finding minimum dimensions for the van (4).
For a more secure Merit, the student would need to calculate and clearly communicate
the correct dimensions of the sail.
© NZQA 2014
Student 3: Low Merit
Calculate Sail dimensions
tan 20 =
x
9
x = tan20 = 9tan 20 = 3.28
x = 3.28m (pole height)
1
AC = (9 2 + 3.28 2 )
AC = 9.58m (Sail side)
2
9.78
= ∠CEA
3
9.78
x=(
) tan −1
3
x = 72.9°
∠CEA = 72.9° (Sail angle)
tan x =
3
Length of pole
3.28 + 2.5 = 5.78m
x
= sin 13
5.78
x = 5.78sin13 = 1.3m
y = (5.78 2 − 1.32 ) = 5.63m
(5.632 − 2.4 2 ) = 5.1m (1dp
Smallest van is 5.1m long
4
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91032
Grade Boundary: High Achieved
4.
For Achieved, the student needs to apply right-angled triangles in solving measurement
problems.
This involves selecting and using a range of methods, demonstrating knowledge of
measurement and geometric concepts and terms and communicating solutions which
would usually require only one or two steps.
This student’s evidence is a response to the TKI assessment resource ‘What’s the
Angle?’
This student has measured at a level of precision appropriate to the task, calculated
the height of the pole (1), the length of one side of the sail (2) and an angle in the sail
(3).
To reach Merit, the student would need to identify the shape and size of the sail and
correctly link the length of the pole to the dimensions of the trailer.
© NZQA 2014
Student 4: High Achieved
T
O A O
C S
A H H
Height of pole
AB = tan23x8 = 3.4m
1
Dimensions of sail
AC = (8 2 + 3.4 2 )
= 75.56
= 8.7 m
2
Triangle ACE
∠CEA = tan −1 (
8.7
)
2
= 77.1°
3
Triangle ABC
x
= tan 13°
3.4
x = 3.4 × tan 13
x = 0.78m
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91032
Grade Boundary: Low Achieved
5.
For Achieved, the student needs to apply right-angled triangles in solving measurement
problems.
This involves selecting and using a range of methods, demonstrating knowledge of
measurement and geometric concepts and terms and communicating solutions which
usually require only one or two steps.
This student’s evidence is a response to the TKI assessment resource ‘What’s the
Angle?’
This student has measured at a level of precision appropriate to the task, calculated
the height of the pole (1) and the length of one side of the sail (2).
For a more secure Achieved, the student would need to more clearly communicate
what the second and third calculations represent.
© NZQA 2014
Student 5: Low Achieved
a)
x
9
tan 21× 9 = x
x = 3.45
= 3.45
tan 21 =
1
The height of the pole is 3.45m
b)
a2 + b2 = c2
9 2 + 3.45 2 = c 2
c 2 = 92.903
c = 9.6m
= 9.6m
c)
9
)
2.1
= 76.9
= 76.9
tan −1 (
2
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91032
Grade Boundary: High Not Achieved
6.
For Achieved, the student needs to apply right-angled triangles in solving measurement
problems.
This involves selecting and using a range of methods, demonstrating knowledge of
measurement and geometric concepts and terms and communicating solutions which
usually require only one or two steps.
This student’s evidence is a response to the TKI assessment resource ‘What’s the
Angle?’
This student has taken some measurements at a level of precision appropriate to the
task, and found the length of one edge of the sail (1). The student has not calculated
the length of AB correctly.
To reach Achieved, the student would need to provide evidence of selecting and using
another method correctly in the solving of a problem.
© NZQA 2014
Student 6: High Not Achieved
AB = sin 24 =
O
H
AB
7.5
7.5 sin 24 = AB
AB = 3.05m(2dp)
=
AC = cos 24 =
A
H
A 7.5
=
H H
7.5
= AC
cos 24
AC = 8.21m(2dp)
1