Numeracy
Tutor Guide
[HIGHER]
Part
Part
Part
Part
Part
Part
Part
1:
2:
3:
4:
5:
6:
7:
National unit specification
Introduction to the unit
Introduction to this pack
Assessment information
Tutor assignments/responses
Attendance/tutor contact requirements
Tutor evaluation form
3
7
9
13
15
23
25
Acknowledgement
Grateful thanks are expressed to SQA for permission to use various extracts from Numeracy
Higher D01C 12 National Unit Specification: General Information
NATIONAL UNIT SPECIFICATION
PART 1
General information
UNIT Numeracy (Higher)
NUMBER D01C 12
Course summary
This unit seeks to develop a wide range of graphical skills in everyday and generalised
contexts and apply in combination a wide range of numerical and statistical skills in
generalised contexts.
Outcomes
1.
Analyse and interpret graphical information.
2.
Select and use appropriate graphical forms to communicate information.
3.
Apply in combination a wide range of numerical and statistical skills.
Recommended entry
While entry is at the discretion of the centre, candidates would normally be expected to
have attained Numeracy (Intermediate 2).
Credit value
1 Credit at Higher.
Core skills
Information on the automatic certification of core skills is published in Automatic Certification
of Core Skills in National Qualifications (SQA, 1999).
The attainment of this unit will lead to the automatic award of:
• Numeracy at Higher.
NUMERACY: TUTOR GUIDE/HIGHER
3
NATIONAL UNIT SPECIFICATION
Statement of standards
UNIT Numeracy (Higher)
Acceptable performance in this unit will be the satisfactory achievement of the standards set
out in this part of the unit specification. All sections of the statement of standards are
mandatory and cannot be altered without reference to the Scottish Qualifications Authority.
Outcome 1
Analyse and interpret graphical information.
Performance criteria
(a)
Extract information accurately from complex diagrams.
(b)
Give full and correct interpretations of significant features.
Note on range for the outcome
Graphical information: may include statistical data in graphical forms.
Significant features: may include patterns; discontinuities; rates of change; turning values;
relationship of variables in, for example, socio-economic data, population growth curves,
predator–prey relationships; graphs of distance/speed/acceleration; sales and stock
movements; weather maps.
Evidence requirements
Oral, written and/or performance evidence that the candidate can analyse and interpret
significant features of at least two pieces of graphical information.
Outcome 2
Select and use appropriate graphical forms to communicate information.
Performance criteria
(a)
Select an appropriate form.
(b)
Use the selected form of communication to present information clearly.
Note on range for the outcome
Communicate information: information communicated in the form of tables; line graphs; bar
charts; pie charts; stem and leaf charts; histograms; diagrams or qualitative forms as
appropriate to the context.
Qualitative form: candidate should be able to recognise relationships in a graph with no
scale on axes, e.g. graph of predator–prey relationship.
Evidence requirements
Evidence that the candidate can select an appropriate form and use the selected form of
communication to present information clearly. At least two pieces of evidence are required.
4
NUMERACY: TUTOR GUIDE/HIGHER
NATIONAL UNIT SPECIFICATION
Outcome 3
Apply in combination a wide range of numerical and statistical skills.
Performance criteria
(a)
Work with a mathematical concept.
(b)
Decide the steps to be carried out.
(c)
Carry out a number of sustained complex calculations.
Note on range for the outcome
Mathematical concept: such as relationship in symbolic forms; or negative numbers (e.g. in
the context of a number line or as temperatures below zero); or the concept of different
types of data (e.g. qualitative, quantitative, discrete, continuous); or statistical concepts (e.g.
standard deviation, confidence limits).
Complex calculations: some examples of complex calculations are: use of formulae in
symbolic form; calculations involving indices (scientific notation); calculation of standard
deviation; manipulation of symbols; addition, subtraction, multiplication and division of
fractions.
Evidence requirements
Oral, written and/or performance evidence that the candidate can:
• solve problems in generalised contexts involving one mathematical concept
• carry out four calculations involving complex sustained calculations in generalised
contexts.
At least half the calculations should involve five steps.
NUMERACY: TUTOR GUIDE/HIGHER
5
6
NUMERACY: TUTOR GUIDE/HIGHER
INTRODUCTION TO THE UNIT
PART 2
What this unit is about
This core skills unit seeks to develop the ability to interpret and also to produce a wide
variety of graphs, charts and other methods of illustrating data as part of the process of
analysing it. It also develops, or perhaps introduces, a wide variety of numerical concepts
or processes which are part of the portfolio of skills required for everyday problem solving
at lower management level posts in business, administration and technician occupations.
Outcomes
1.
Analyse and interpret graphical forms
2.
Select and use graphical forms
3.
Apply a wide range of numerical and statistical skills
Prior experience
While entry is at the discretion of the centre, candidates would normally be expected to
have attained Numeracy (Intermediate 2).
Progression
At time of publication this is the highest level of Numeracy which is assessed by SQA.
Core skills
Information on the automatic certification of core skills is published in Automatic Certification
of Core Skills in National Qualifications (SQA, 1999).
The attainment of this unit will lead to the automatic award of
• Numeracy at Higher.
NUMERACY: TUTOR GUIDE/HIGHER
7
8
NUMERACY: TUTOR GUIDE/HIGHER
INTRODUCTION TO THIS PACK
PART 3
This open learning pack covers the syllabus requirements for the SQA National Unit,
Numeracy (Higher). In addition to this Tutor Guide, it consists of a Student Introductory
Guide and two Study Sections to cover the three outcomes, as indicated below.
Student Introductory Guide
Pages: 14
Study time: 1 hour
Outcome 1/2: Interpreting Graphical Information and
Using Graphical Information
Pages: 54
Study time: 15-20 hours
Outcome 3: Numerical Skills
Pages: 143
Study time: 20-25 hours
Unit Study Sections
Each Study Section of this learning pack contains the following:
• Contents page
• An introduction to the section
– what the section is about
– the objectives of the section
– other resources required which are not included in the package
• Assessment information
– this will have to be communicated by the learning provider
• Subject content, including
– Self Assessed Questions (SAQs) and Tutor Assignments (TAs)
– answers to questions
– comments on Activities
NUMERACY: TUTOR GUIDE/HIGHER
9
INTRODUCTION TO THIS PACK
It is as well at this stage to make the following observation about Outcome 3. The content
suggested by the unit specification for Higher is extremely wide ranging. The writer has
chosen to pick several topics and explore them in depth. These have been organised into
five components. As the evidence requirements stipulate four questions from sufficiently
different areas of study, the student should be directed to the four topics which you feel are
most appropriate.
Other topics may be added at your discretion; or you may wish to augment or replace some
of the existing topics.
Remember that, if you do this, SAQs and TAs may have to be amended too.
Approximate study time
The notional design length for the course is 40 hours, but this will depend very much on the
academic level and amount of time committed to studying by the student. The study times
quoted should be used as a very rough guide.
Revision
Depending on their background, you may find it necessary to provide extra support for
some students, particularly in the area of graph drawing.
The corresponding pack at Intermediate 2 should provide sufficient background and
practice. You can photocopy relevant pages and issue them to students at your discretion.
Note on standard deviation
In the section dealing with this (Outcome 3, Section 1) the author has used the true
standard deviation formula with n rather than the adjusted formula with n–1. Depending on
their previous experience, some students may query this.
Activities
An activity is a short piece of work for a student to carry out or think about. It is generally
less demanding than an SAQ. Activities should be attempted (and answers/comments
checked at the back) before students carry on with the study section.
10
NUMERACY: TUTOR GUIDE/HIGHER
INTRODUCTION TO THIS PACK
Symbols used in the pack
In addition to the Introductory Guide there are three distinct Study Sections, each relating
to a unit outcome. The Study Sections allow students to work on their own with your
support. As they work through the sections, they will encounter two symbols which
indicate that something follows that they are expected to do. An explanation of these
symbols follows:
Self Assessed Questions
?1
This symbol is used to indicate a numbered Self Assessed Question (SAQ). SAQs are used
most commonly to check the students’ understanding of the material that has already been
covered in the Study Sections.
This type of assessment is self-contained, that is to say that everything is provided within
the Study Section to enable students to check their understanding of the materials.
The process is simple:
• The students are set SAQs throughout the Study Section. These will be set as e.g. shortanswer questions, etc.
• They respond to these, by writing in their own notebook.
• On completion of the SAQ, they turn to the back of the section to compare the SAQ
responses to theirs.
• If they are not satisfied after checking out their responses, they should turn to the
appropriate part of the Study Section and go over the topic again.
Remember – the answers to SAQs are contained within the study materials. Students are
not expected to ‘guess’ at these answers.
Tutor Assignment
T1
This symbol is used to indicate a Tutor Assignment. These are usually found at the end of
each Study Section. The aim of the TA is to cover and/or incorporate the main topics of the
Study Section and prepare the student for unit (summative) outcome assessment.
Tutors may want to change individual questions in the TAs to reflect more closely the
questions set in the assessments used in their own teaching establishments.
NUMERACY: TUTOR GUIDE/HIGHER
11
12
NUMERACY: TUTOR GUIDE/HIGHER
ASSESSMENT INFORMATION
PART 4
How students will be assessed
Throughout each Study Section of this learning package, a series of Self Assessed Questions
(SAQs) have been developed to provide students with ‘on-the-spot’ feedback about their
progress within the relevant Section.
Upon successful completion of all SAQs, students will be asked to attempt a Tutor
Assignment (TA). Each Section usually finishes with a TA and each assignment has been
devised as a means of assessing the student’s progress on the knowledge and understanding
required for their SQA unit to date. Generally, the requirements of the TAs closely match
the outcomes of the unit.
When and where students will be assessed
As a tutor, you should summatively assess each student after successful completion of the
appropriate TA, using your own centre’s instrument(s) of summative assessment.
Depending on the policy of your school or college, summative assessment may be
undertaken at the centre, or at a distance from the centre, under supervision.
Most often assessment is undertaken by the learner at school or college under supervision of
a tutor. However, for the student who lives some distance from the school or college, an
invigilation system may be set up at a recognised support centre local to the student
(community education centre, training centre, etc).
The summative assessments are recorded by you, the tutor, and they form the basis of the
student’s final results within the unit. The student should be informed that you will
complete all the necessary paperwork and notify them of their successful completion of the
unit.
What students have to achieve
All outcomes have to be assessed and the objective of this 40-hour unit is that the student
will achieve Outcomes 1, 2 and 3 of the SQA unit Numeracy (Higher) D01C 12.
NUMERACY: TUTOR GUIDE/HIGHER
13
ASSESSMENT INFORMATION
Opportunities for reassessment
If students don’t achieve the required standard for the award of ‘pass’ within any
assessment, they should be informed of this and you should arrange for them to be
reassessed on the particular elements of the assessment which needs improving.
Alternative instruments of summative assessment should be available and utilised where
necessary.
External assessment
There is no external assessment associated with this unit.
14
NUMERACY: TUTOR GUIDE/HIGHER
TUTOR ASSIGNMENTS/RESPONSES
PART 5
Answers to Outcome 1/2 Tutor Assignments
T1: Answers
The graphs to be drawn are not illustrated here, for obvious reasons. Because of the wide
variety of scales and styles, tutors will need to check each individual illustration for
appropriateness and accuracy.
Suggested answers might be:
1.
2.
3.
4.
5.
compound bar chart
ordinary line graph
line graph with two lines
component bar chart, using either raw figures or percentages
pie chart.
NB: Variations in numerical answers will occur from now on. They will depend on the
readings taken, which are all approximate.
T2: Answers
(a)
1985: £94.2 million ÷ 57 million journeys = £1.65
1986: £99.5 million ÷ 53 million journeys = £1.88
23p increase
(b)
Air journeys show a steady increase.
Rail journeys are more erratic, a general decrease 1985–89 with a (possible)
recovery in 1990.
T3: Answers
(a)
(b)
(c)
1985: £12.2 million
1987: Pie chart drama angle 72° representing 20%
1987: £13.9 million spent
Literature angle 17°
Cash spent:
17
× £13.9m = £656,000
360
NUMERACY: TUTOR GUIDE/HIGHER
15
TUTOR ASSIGNMENTS/RESPONSES
T4: Answers
17–20:
20–30:
30–40:
40–50:
50–65:
10,000
15,000
25,000
22,000
12,000
84,000
12%
18%
30%
26%
14%
100%
T5: Answers
(a)
Wages just under 55%
Materials about 13%
Overheads about 35%
(b)
Wages 12% of £190,000 = £22,800
Materials 30% of £190,000 = £57,000
Overheads 58% of £190,000 = £110,200
(c)
Wages costs are very high (around 90%)
Materials and overheads are very low (5% each)
Answers to Outcome 3 Tutor Assignments
T1: Answers
Time (min)
10.0 –10.4
10.5 –10.9
11.0 –11.4
11.5 –11.9
12.0 –12.4
12.5 –12.9
13.0 –13.4
x=
f
5
12
23
10
5
2
1
58
x
10.2
10.7
11.2
11.7
12.2
12.7
13.2
653.6
= 11.27 mins
58
7,389.12  653.6 
–

58
 58 
= 0.64 mins
2
s=
16
NUMERACY: TUTOR GUIDE/HIGHER
fx
51
128.4
257.6
117
61
25.4
13.2
653.6
fx2
520.2
1,373.88
2,885.12
1,368.9
744.2
322.58
174.24
7,389.12
TUTOR ASSIGNMENTS/RESPONSES
T2: Answers
1.
19
22
22
24
24
V
26
27
30
31
V
33
34
40
45
V
Q1 = 23, Q 2 = 27, Q3 = 33.5
SIQR = 5.25
2.
3.
x
0
1
2
3
4
5
6
Interval
25-30
30-35
35-40
40-45
45-50
50-55
f
47
53
20
8
5
3
1
137
c.f.
47
100
120
128
133
136
137
f
1
14
23
21
15
6
80
c.f.
1
15
38
59
74
80
Rank of Q 1 = 35
Rank of Q 2 = 69
Rank of Q3 = 103
So
Q1 = 0 faulty bulbs
Q2 = 1 faulty bulb
Q3 = 2 faulty bulbs
SIQR = 1 bulb
Rank of Q1 = 20
Rank of Q2 = 40
Rank of Q3 = 60
20 – 15
× 5 = 36.1 i.e. £36,100
23
40 – 38
Q2 = 40 +
× 5 = 40.5 i.e. £40,500
21
60 – 59
Q3 = 45 +
× 5 = 45.3 i.e. £45,300
15
Q1 = 35 +
T3: Answers
1.
2.
3.
4.
5.
3Wc
;
b
2P + c
;
a=
2b
5 – Aq
;
p=
q
R – 6πm
;
t=
3π
y (z 2 + 16)
;
x=
16
a=
c=
ab
3W
c = 2(ab – P )
5
A+p
3πt – R
m=
6π
16x
y= 2
(z + 16)
q=
NUMERACY: TUTOR GUIDE/HIGHER
17
TUTOR ASSIGNMENTS/RESPONSES
T4: Answers
Parts (a) and (d)
18
NUMERACY: TUTOR GUIDE/HIGHER
TUTOR ASSIGNMENTS/RESPONSES
(b)
Authority
A
B
C
D
E
F
G
H
I
J
(c)
r=
Population
x (thousand)
145
236
95
142
178
263
74
137
89
312
1,671
Amount
y (£m)
7.2
9.5
3.6
4.2
9.3
10.1
3.9
5.6
3.3
11.4
68.1
x2
21,025
55,696
9,025
20,164
31,684
69,169
5,476
18,769
7,921
97,344
336,273
y2
51.84
90.25
12.96
17.64
86.49
102.01
15.21
31.36
10.89
129.96
548.61
xy
1,044
2,242
342
596.4
1,655.4
2,656.3
288.6
767.2
293.7
3,556.8
13,442.4
10 × 13,442.4 – 1.671 × 68.1
[10 × 336,273 – 1,6712 ] × [10 × 548.61 – 68.12 ]
20,628.9
= 0.93762 ... = 0.94 (to 2 significant figures)
570,489 × 848.49
10 × 13,442.4 – 1,671 × 68.1
b=
10 × 336,273 – 1,6712
20,628.9
=
= 0.03616 ... = 0.036 (to 2 significant figures)
570,489
68.1 – 0.03616 × 1,671
a=
10
= 0.76765 ...
=
= 0.77 (to 2 significant figures)
Line is y = 0.77 + 0.036x
(e)
(f)
(i)
x = 153, y = 0.77 + 0.036 × 153
= 6.278
i.e. approx. £6.3 million
(ii)
x = 406, y = 0.77 + 0.036 × 406
= 15.386
i.e. approx. £15.4 million
Value of correlation coefficient close to 1 shows very strong relationship. The £6.3
million estimate should be reliable because of this, also because the population of
£153,000 is well within the known range of data. The other estimate is for a
population well outwith the known range of data, so is not expected to be so reliable.
NUMERACY: TUTOR GUIDE/HIGHER
19
TUTOR ASSIGNMENTS/RESPONSES
T5: Answers
1.
(a)
Un = (1.15)n U0 where U0 = initial value £1m, U1 = value at end of first year.
(i)
1.153 × £1m = £1,520,875
(ii)
1.155 × £1m = £2,011,357
(b)
Un+1 = 1.15 Un
U7 = £2,660,020
U8 = £3,059,023
i.e. trebles in value after 8 years.
2.
(a)
Month
Nov
Dec
Jan
Feb
Mar
Apr
May
Jun
Jul
Owing at start
£4,000
£3,580
£3,151
£2,714
£2,268
£1,814
£1,350
£877
£395
Owing at end
£4,080
£3,651 (omitting pennies)
£3,214
£2,768
£2,314
£1,850
£1,377
£895
£403
Paid back at the start of August, only £403.06 being required, not the full £500.
3.
(b)
Un+1 = 1.02 Un – 500 with U0 = £4000
The eighth press of the = key gives £395.152999.
The ninth press gives –£96.94 showing the loan has been (more than) paid
off.
(a)
(b)
(c)
Un+1 = 1.08 U n – 1.75, U0 = 20 (numbers refer to millions)
After four years there are 19,324,083 fish.
Looks good for the fish, but if you look at the sequence
20
←
0.15 difference
←
0.162 difference
←
0.175 difference
←
0.191 difference
19.85
19.688
19.513
19.324
20
NUMERACY: TUTOR GUIDE/HIGHER
TUTOR ASSIGNMENTS/RESPONSES
you see that the differences are increasing, i.e. the sequence is diverging, so there is
no limit. (1.08 is outside the –1 to +1 range required for convergence.) In fact there
will be no fish after 32 years. The permitted arrival catch is too large.
Note
6
gives the answer 21.875 which makes no sense at all in the
1–a
context of the question.
The limit formula
4.
(a)
Un+1 = 0.80 Un + 2,000
(b)
L=
2,000
= 10,000
1 – 0.8
i.e. number of turkeys varies between 10,000 immediately before Christmas
and 8,000 immediately after. Not a very economic policy!
NUMERACY: TUTOR GUIDE/HIGHER
21
22
NUMERACY: TUTOR GUIDE/HIGHER
ATTENDANCE/TUTOR CONTACT REQUIREMENTS
PART 6
When students enrol for this course they should either be given a timetable or receive
details of their tutor and information on contact details, i.e. the day, time, telephone/e-mail
number, where they can make contact. They may retain this information on a Tutor Details
Form similar to that shown below.
Tutor Details Form
Tutor’s name:
Address (for correspondence and assignments):
Telephone number:
Fax number:
E-mail address:
Times available for contact:
Day/Evening:
Times:
Attendance requirement:
NUMERACY: TUTOR GUIDE/HIGHER
23
24
NUMERACY: TUTOR GUIDE/HIGHER
TUTOR EVALUATION FORM
PART 7
Learning and Teaching Scotland is very interested in the views of tutors who have used
these learning materials with students. Your feedback and comments will assist us in
evaluating and, where necessary, improving this package for future student and tutor use.
We would be grateful if you would spend a little time completing and returning this form to
Learning and Teaching Scotland.
Please answer all of the questions as fully and frankly as possible. Please rate the materials
by placing a tick in the appropriate box and adding relevant comments in the space
provided.
Thank you for your assistance.
1
The content is pitched at the appropriate
level for the target student
Agree
Strongly
Agree
Disagree Disagree
Strongly
2
The content is accurate and up-to-date
Agree
Strongly
Agree
Disagree Disagree
Strongly
3
The content meets the requirements of
the stated outcomes/aims/objectives
Agree
Strongly
Agree
Disagree Disagree
Strongly
4
The content is sufficient to allow the student Agree
to achieve the stated outcomes/aims/
Strongly
objectives
Agree
Disagree Disagree
Strongly
5
The learning approaches are appropriate
Agree
Disagree Disagree
Strongly
Agree
Strongly
contd overleaf
NUMERACY: TUTOR GUIDE/HIGHER
25
TUTOR EVALUATION FORM
6
The language is suitable for the target student Agree
Strongly
Agree
Disagree Disagree
Strongly
7
Sufficient and significant exercises, SAQs
and Tutor Assignments are included
Agree
Strongly
Agree
Disagree Disagree
Strongly
8
Appropriate feedback has been included
Agree
Strongly
Agree
Disagree Disagree
Strongly
9
The assessment methods are appropriate
Agree
Strongly
Agree
Disagree Disagree
Strongly
10
The standards of assessment are acceptable
Agree
Strongly
Agree
Disagree Disagree
Strongly
11
The pack is structured in such a way as
to allow students to find their way through
the materials
Agree
Strongly
Agree
Disagree Disagree
Strongly
12
This pack is appropriate for use with a
minimum of tutor contact
Agree
Strongly
Agree
Disagree Disagree
Strongly
13
Overall I would rate this pack as
Very
Good
Good
Name
School/College
Poor
Very
Poor
Date
Thank you once again for your assistance. Please send completed forms to: OFDL Project,
Learning and Teaching Scotland, Gardyne Road, Dundee DD5 1NY
26
NUMERACY: TUTOR GUIDE/HIGHER