Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91038
Exemplar for Internal Achievement Standard
Mathematics and Statistics Level 1
This exemplar supports assessment against:
Achievement Standard 91038
Investigate a situation involving elements of chance
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
91038
Grade Boundary: Low Excellence
1.
For Excellence, the student needs to investigate, showing statistical insight, a situation
involving elements of chance.
This involves integrating contextual information and knowledge with an understanding
of applications of probability and may involve considering the possible effects of other
related variables or factors.
This student’s evidence is a response to the TKI assessment resource ‘Games of
Chance’.
This student has used the experimental probability process to investigate a situation.
The student has posed a question (1), planned an experiment (2), gathered data (3),
selected and used appropriate displays including the experimental probability
distribution (4) and answered the question (5).
The student has integrated contextual information and knowledge by identifying
patterns, giving an expected result and answering the question in context (6). The
student has demonstrated an understanding of applications of probability by
considering the theoretical probability of getting five sixes and relating this to the
experimental situation (7).
For a more secure Excellence, the student could have discussed the shape of the
probability distribution and the theoretical probability in greater depth.
© NZQA 2014
I am going to investigate the number of sixes I will get when I roll 5 dice. I know I will get either 0,
1, 2, 3, 4 or 5 sixes.
1
I think I will probably get mostly one six or two sixes.
I am going to roll five dice together 50 times onto my desk so that I keep the conditions the same I
am going to count the number of sixes each time.
Number of sixes
0
1
2
3
4
5
|||| |||| |||| ||
|||| |||| |||| |||| |||
|||| ||||
|
Number of times
17
23
9
1
0
0
2
3
4
Looking at my bar graph I can see 1 six is the most common result and then 0. This happened 40
times which is 80%. In the long run I would expect to get 0 or 1 six most of the time. I didn't get 4 or
5 sixes, but both of these are possible results.
I thought I would get mostly 1 or 2 sixes but I actually got mostly 0 or 1 six. I noticed that I got a lot
of 4s when I threw the dice so if I was doing 4's my results may have been different. The chances
of getting a 4 and 6 would be the same if the dice were fair so it shouldn't matter.
I can see why I didn't get any of 5 6's because there is 1 chance in 6 of getting a six and
if I multiply that together 5 times I will get a very small number.
7
5
6
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91038
Grade Boundary: High Merit
2.
For Merit, the student needs to investigate, with justification, a situation involving
elements of chance.
This involves linking aspects of the investigation to the situation, and making
supporting statements which refer to evidence such as summary statistics,
probabilities, trends or features of visual displays.
This student’s evidence is a response to the TKI assessment resource ‘Games of
Chance’.
The student has used the experimental probability process to investigate a situation.
The student has posed a question (1), planned an experiment (2), gathered data (3),
selected and used appropriate displays (4), found experimental probabilities (5) and
answered the question (5).
The student has linked aspects of the investigation to the situation by making
supporting statements in context which refer to patterns in the data, giving an expected
result and answering the question (6).
To be awarded Excellence, the student would need to provide more detail and a
greater contextual understanding in the expected result and in their discussion on the
long run relative frequency.
© NZQA 2014
I am going to investigate the chances of getting four of a kind in Yahtzee.
1
In each turn I can roll the 5 dice three times and keep as many dice as I want. Initially I will do this
10 times and record a tick if I get 4 of a kind and a cross if I don't and then count the number I get.
I will then repeat this for some more sets of 10 so that I have more results to see a pattern.
2
I think I will get four of a kind about a quarter of the time.
Turn
4 of a kind
1

2

3

4

5

6

7

8

9

10

18

28

38

48

58

68

78

88

98

19

29

39

49

59

69

79

89

99

20

30

40

50

60

70

80

90

100

1
After 10 turns I got four of a kind once so the probability is 0.1.
Turn
4 of a kind
Turn
4 of a kind
Turn
4 of a kind
Turn
4 of a kind
Turn
4 of a kind
Turn
4 of a kind
Turn
4 of a kind
Turn
4 of a kind
Turn
4 of a kind
11

21

31

41

51

61

71

81

91

12

22

32

42

52

62

72

82

92

13

23

33

43

53

63

73

83

93

14

24

34

44

54

64

74

84

94

15

25

35

45

55

65

75

85

95

16

26

36

46

56

66

76

86

96

17

27

37

47

57

67

77

87

97

0
2
1
2
3
1
4
1
0
2
Summary
After 10
After 20
After 30
After 40
After 50
After 60
After 70
After 80
After 90
After 100
1 time
1 time
3 times
4 times
6 times
7 times
11 times
12 times
12 times
14 times
4
The number of four of a kind I got varies but the most was 4 in a set of 10 and the lowest was
none. The summary of results is what happened when I did more trials and the graph shows how
the probability changed with more trials.
I noticed that the peak in the long run frequency graph occurred in the 7th lot of 10 when I got four
of a kind four times. After 100 trials I had four of a kind 14 times which is a probability of 0.14.
After 100 trials I didn’t get four of a kind 86 times which is a probability of 0.86.
6
5
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91038
Grade Boundary: Low Merit
3.
For Merit, the student needs to investigate, with justification, a situation involving
elements of chance.
This involves linking aspects of the investigation to the situation and making supporting
statements which refer to evidence such as summary statistics, probabilities, trends or
features of visual displays.
This student’s evidence is a response to the TKI assessment resource ‘Games of
Chance’.
The student has used the experimental probability process to investigate a situation.
The student has posed a question (1), planned an experiment (2), gathered data (3),
selected and used appropriate displays including the experimental probability
distribution (4), identified and communicated a pattern in the data (5) and answered the
question (6).
The student has linked aspects of the investigation to the situation by making
supporting statements in context and answering the question (6).
For a more secure Merit the student would need to provide more detail in
communicating findings. For example, the discussion of the skew in the distribution
could be developed further, and the theoretical probability investigated.
© NZQA 2014
I am going to investigate the number of 3's I expect to get when I roll 5 dice.
1
I am going to roll the 5 dice 50 times. Each time I will count how many of the dice are a 3. I can get
0 – 5 3's.
2
On average I think I will get less than one 3 each turn because there are five dice and there is a 1
in 6 chance of getting a 3.
Number of 3’s
0
1
2
3
4
5
|||| |||| ||||
|||| |||| |||| |||| |
|||| |||| |
||
|
Frequency
15
21
11
2
1
0
Probability
0.30
0.42
0.22
0.04
0.02
0.00
3
4
The most common result I got was one 3 and this happened 21 times which is 0.42.
The shape of the graph is skewed with high numbers at the beginning and then low numbers at the
end. Most of the time there will be zero or one 3. There were 14 times, which is a probability of
0.28, when I didn't get zero or one 3.
6
5
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91038
Grade Boundary: High Achieved
4.
For Achieved, the student needs to investigate a situation involving elements of
chance.
This involves using the experimental probability process.
This student’s evidence is a response to the TKI assessment resource ‘Games of
Chance’.
The student has used the experimental probability process to investigate a situation.
The student has posed a question (1), planned an experiment (2), gathered data (3),
selected and used appropriate displays including the experimental probability
distribution (4) and answered the question (5).
The student has also identified and communicated patterns in the data (6).
To be awarded Merit, the student would need to justify comments with supporting
evidence from the experiment.
© NZQA 2014
1
What is the probability of getting at least one double when 5 dice are rolled?
I am going to roll 5 dice together and count the number of doubles of any number. For example if
the numbers are 5, 6, 3, 5, 2, I will count one double. If the numbers are 3, 4, 1, 4, 3, I will count
two doubles. If the numbers are 4,5,6,2,1 it is 0 doubles. If there are 3 or 4 numbers the same I will
only count this as one double. I will roll the dice 50 times.
I reckon that two doubles will come up most times.
Number of
doubles
0
1
2
Number of
times
4
35
11
Probability
3
4/50
35/50
11/50
4
The probability of getting at least one double is 92%.
5
The most common number was 1 and this happened 70% of the time.
There was nearly three times as many two doubles as no doubles.
6
2
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91038
Grade Boundary: Low Achieved
5.
For Achieved, the student needs to investigate a situation involving elements of
chance.
This involves using the experimental probability process.
This student’s evidence is a response to the TKI assessment resource ‘Games of
Chance’.
The student has used the experimental probability process to investigate a situation.
The student has posed a question (1), planned an experiment (2), gathered data (3),
selected and used appropriate displays including the experimental probability
distribution (4) and answered the question (5).
For a more secure Achieved, the student would need to provide a more detailed
answer to the question and draw the graph of the experimental distribution accurately.
© NZQA 2014
Question
How many sixes would I expect to get when five dice are rolled.
1
Plan
I am going to roll five dice together and count the number of sixes.
2
I am going to roll the five dice 50 times.
The number of sixes I can get could be all of the dice showing a six, none of them showing six and
any number in between.
Number of sixes
0
1
2
3
4
5
Number of times
21
20
6
2
1
0
|||| |||| |||| |||| |
|||| |||| |||| ||||
|||| |
||
|
Probability
0.42
0.40
0.12
0.04
0.02
0.00
3
4
Number of times
Experiment results
25
20
15
10
5
0
1
2
3
4
5
6
Number of sixes
The most common number was 0.
The larger numbers happen less times than the smaller numbers.
I would expect to get 0 or 1 six when I toss five dice.
6
5
Exemplar for internal assessment resource Mathematics and Statistics for Achievement Standard
91038
Grade Boundary: High Not Achieved
6.
For Achieved, the student needs to investigate a situation involving elements of
chance.
This involves using the experimental probability process.
This student’s evidence is a response to the TKI assessment resource ‘Games of
Chance’.
The student has posed a question (1), planned an experiment (2), gathered data (3),
selected and used an appropriate display (4), identified and communicated patterns in
the data (5) and answered the question (5).
There is insufficient evidence that the student used the experimental probability
process to investigate a situation.
To be awarded Achieved, the student would need to give the experimental probability
distribution and identify a further pattern in the data.
© NZQA 2014
1
What are the chances of getting a double with five dice?
2
I am going to roll five dice fifty times and count the number of doubles of any number.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
5
3
4
2
3
2
1
1
2
2
4
2
5
6
4
2
6
2
3
5
6
4
5
2
6
1
3
4
4
3
4
2
3
4
3
1
1
6
3
4
1
3
5
3
3
1
5
2
5
5
1
3
6
6
2
6
5
6
6
2
2
6
5
4
4
6
5
6
6
2
3
1
4
1
5
2
4
1
6
2
5
3
1
5
2
2
6
1
1
1
2
4
1
3
3
5
4
3
6
3
3
4
3
5
1
6
4
6
3
4
4
6
2
4
6
1
5
2
2
6
2
4
3
4
2
1
2
1
1
2
1
0
2
0
1
2
1
1
1
1
2
1
1
1
1
0
1
1
0
1
Number of
doubles
0
1
2
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
Number of
times
6
33
11
Sometimes I got a triple and I counted this as a double.
My chances of getting a double are 33/50 = 66%.
2
2
1
3
2
6
1
3
2
4
6
2
5
2
3
2
5
3
4
3
2
5
4
3
2
5
6
1
4
6
5
1
5
6
3
5
1
4
4
2
4
2
4
1
6
6
4
3
1
6
4
6
5
5
1
5
3
1
1
4
4
4
4
6
2
5
2
4
1
5
6
1
2
2
6
4
4
2
6
4
4
6
4
6
3
5
1
2
6
5
2
4
1
6
2
5
2
2
2
2
3
4
2
2
5
5
4
6
5
1
2
4
4
5
4
6
1
2
2
5
6
6
4
6
3
6
1
1
1
1
1
0
1
2
1
2
2
1
1
2
1
1
2
1
1
1
0
1
1
1
2
1
3
4