PST201F/201/1/2019
Tutorial Letter 201/1/2019
MATHEMATICS AND MATHEMATICS TEACHING
PST201F
Semesters 1
Department of Mathematics Education
This tutorial letter contains important information
about your module.
BARCODE
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1.1
1.2.
The question is subjective depending on the individual experiences.
Picture 2: Students are engaged in mathematics classroom using some manipulatives to
respond to the task.
Picture 6
The teacher is visible to learners – presenting a lesson in front of the classroom and building
new knowledge from prior knowledge. The learners’ interest to observe closely is aroused
and the background wall displays tools and manipulatives for mathematical concepts. The
teacher is standing in front, explaining and demonstrating to learners who are
enthusiastically listening to the teacher and actively involved in the lesson.
1.3
Roles of teacher:
a. Facilitate learning in the classroom.
b. Creates learning environment where all learners are able to learn without fear.
c. Provides opportunity for learners to learn with understanding
Roles of learners:
a. Share their ideas.
b. Listen to each other.
c. Look for and discuss connections.
1.6
Assimilation: It occurs when new concepts fit into an existing network of ideas. The new
information expands the pre-existing network. It refers to the use of an existing schema to
give meaning to new experiences. Assimilation is based on learners' abilities to notice
similarities among objects and match new ideas to those they already possess.
Accommodation: It takes place when new concepts do not fit into an existing network of
ideas. The brain has to revamp or re-organise the network to accommodate the new
concepts.
Disequilibrium: When new knowledge and pre-existing knowledge do not match and there
is a need to modify the rearrangement of concepts and connections to accommodate the
new knowledge.
Reflective thought: A sift through the pre-existing ideas in order to find those that seem
related to the new knowledge and how they are related.
1.7
a. Ronny has used the concept of equal sharing to solve the problem, While Michael has
decided to take concept of multiples of 4 in solving the problem.
b. Algorithm is systematic way of doing certain procedures to arrive at acceptable answer.
1.9
Seven weaknesses of rote learning:

Learners do not always understand what they are learning.

They will not enjoy the subject.

They may perform poorly.

They forget what is taught, often because there are too many rules and algorithms to
memorize.
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1.11
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
They are not able to apply their knowledge to a new situation.

They do not know how this knowledge applies or links to real to the world.

They cannot develop nor connect concepts.
Relational understanding is understanding of what to do and why. All concepts are in a
connected network. When a learner wants to recall a concept, all other related concepts are
understood and can easily be recalled. Instrumental understanding is on the opposite side
of the continuum of understanding. It is called “doing without understanding”. Ideas are
isolated and cannot be connected
1.13 Dienes blocks : Model base 10 concept.
Spinners
: Model chance (probability).
Thermometer : Illustrates positive and negative Integers.
Rods
: Demonstrate the measure of length.
Geoboards
: Model the property of 2 dimensional shapes.
1.14 Strategic competence: It is the ability to formulate, represent and solve mathematics
problems.
Conceptual understanding: It is comprehension of mathematical concepts, operations and
relations.
Procedural fluency: It is a skill in carrying out procedures flexibly, accurately, efficiently and
appropriately.
Adaptive reasoning: It is capacity for logical thought, reflection, explanation and
justification.
Productive disposition: It regarded as habitual inclination to see mathematics as sensible,
useful and worthwhile, coupled with a belief in diligence and one's own efficacy.
2.1
Teaching for problem solving: teaching skills that students can use later to solve problems.
Teaching about problem solving: teaching students on how to solve problems giving them
general strategies.
Teaching through problem solving: learning mathematics concepts through solving
problems.
2.6
Develop a task and use variety of methods to solve the problem e.g Find the sum 23 and
38
Example: Compensation method
Re-arranging principle
23 +38 = (23-3) + (38+2)+1
23 + 38 = 20 + 3 + 30 + 8
=20+40+1
=(20 + 30) + 3 + 8
=61
=(20 + 30) + 11
=(20 +30 +10) + 1
=61
2.7
Before: Getting ready, activate prior knowledge, be sure the problem is understood, and
establish clear understanding.
During: Teacher instruct students to work, encourages students’ mathematical thinking,
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provides support and guidance, and provides worthwhile opportunity extensions.
After: Class discussion, Teacher promotes mathematics learning communities, listens
actively and summaries main ideas.
PART C
1
2
Counting by one: Counting each piece on items.
Counting by groups of tens and ones: Count a group of ten as single item
Non-Standard base ten: Group the pieces flexibly including tens and ones.
Learners should use explicit base- ten language rather than using standard number
words. Example 5 tens and 6 ones instead of "fifty-six" which for others could mean
506.
3(a)
exchange 10 tinnies into 1 long
=
(b)
1 flat , 8 longs and 5 tinnies
185
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exchange 1 long
into 10 tinnies and the total will then be 11 tinies
Cross out 7 tinnies
and remain with 4 tinnies.
Exchange 1 flat to 10 longs and cross out six longs
= 4longs and 4 tinnies
(c)
44
Exchange 1 long into 10 tinnies.
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
16 tinnies is 4 groups of 4 tinnies
Then exchange 2 flats into 20 longs
24 longs is the same as 240 tinnies
Then 60 groups of 4 tinnies plus 4 groups of 4 tinnies =64 groups of tinnies which is the
same as 6 longs and 4 tinnies
64
4
5.1
A number is divisible by 9 if the sum of all the digits is a multiple of 9
4+0+2+5+3+7+5+9+8+1 = 44
4 025 375 981 is not divisible by 9 because 44 is not a multiple of 9.
To test if the number is divisible 11:
Add every second digit and write the sum, add the remaining digits and write the sum.
Then find the difference between the two sums. The difference should either be 0 or
multiple of 11.
4-0+2-5+3-7+5-9+8-1 = 0
Therefore 4 025 375 981 is divisible by 11.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
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All unshaded numbers are prime numbers
5.2
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
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
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Count off 13; 26; 39;52;65;78 and 91. Therefore the answer is 7 remainder 5.
5.3
240
2
120
2
60
2
2
2
2
2
3
5
30
2
15
3
5
240
120
60
30
15
5
1
2  2  2  2  3  5  24  3  5
6
Horizontal algorithms
a) 451 +196
(400+50+1)+(100+90+6)
=(400+100)+(50+90)+(1+6)
=(500+100)+40+7
=647 (Group the hundreds, tens and the units together)
b) b) 467 -288
(400+60+7) - (200+80+8)
=400+ 60 +7 -200 -80 -8
= 300 + (100 + 50)+(10+7)-200-80-8
=300+150+17-200-80-8
=(300-200)+(150-80)+(17-8)
=100+70+9
=179 (8 units is more than 7 units so we borrow 10 units from the tens to make 17
units.
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Vertical algorithms
H
T
U
1
5
1
4
+1
9
6
6
4
7
5 tens and 9 tens makes 14 tens the 10 tens is same as 100 it should be carried to
the hundreths column.
H
T
U
1
1
43
6 5
7
-2
8
8
1
7
9
8 units is more than 7 units so we borrow 10 units from the tens to make 17 units.
7
8
567 - 329
= 567 + 1 - (329+1)
= 568 - 330
= 238
9
Add 1 to 329 to make it 330 we also add 1 to 567 as we will be subtracting 330 no longer
329.
This misconception can be overcomed by introducing vertical column place value
method.
Tth
6
Th
7
H
5
T U
7 6
Then the underlined digit is shown to be in the Ten thousand Column then its value
must be sixty thousand (60 000).
10
3 items arranged into 4 rows or 4 rows of three or 4 groups of three
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4 items arranged into 3 rows OR 3 rows of 4 or three groups of 4
Let the number of shelves be x.
365x  449680
11
365x 449680

365
365
x = 1 232. It means that 1 232 shelves will be needed
PART D
1. Give examples of each of three categories of fractions. [Illustrate it using diagrams.]
1.1
Area model:
The shaded part represents one part out of four parts
NB: The fractional parts must be equal
1.2 Set model of fraction
OR
Mother shares 20 strawberries
among her 4 children
Nine of the fifteen counters are
9
purple or
are purple
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1.3 Length model
2
In the context of choosing a “whole”, explain when a “fifth” is not always equal to a
“fifth”. Give an example.
When thinking of a fraction as a part of a whole, the part will depend on the size of
the whole. For example, one fifth of an apple is not the same as one fifth of a peach. So
when we contextualise fractions as a part of "something", this "something" has to be
exactly the same. In the world of numbers, when we talk about a fifth, we always mean
a fifth of ONE. (One unit)
3.1 If the given circle represents one whole, then show

Three thirds

One tenth

Five halves
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3.2 If 18 counters are a whole set, how many are there in one third of the set? Illustrate it using an
example.
One third of a set of 18 counters is 6 counters (green)
     
     
     
3.3 If 12 counters are three quarters of a set, how many counters are in a full set?
Illustrate using an example.


3 quarters
The whole set


1 quarter
Full set is equal to 16 counters
4
Bengu make the following sketches to compare the fractions
diagrams set out below. He then states that
3
3
and
5
4
by drawing the
3 3
=
. Discuss Bengu’s misconception and
4 5
illustrate how you will rectify it.
Bengu should be aware of the following when comparing fractions:

Compare using the same whole

Use a fraction wall or the lowest common denominator for comparison

Make a conclusion after comparison
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It is clear from the diagrams that
5. What is the fractional relationship between
is one third of
and
, hexagon divided into three equal parts. One part is a third.
6. Use the following models to show the equivalence of the fractions
6.1 A set model
6.2 An area model
? Illustrate.
and
:
13
7. Compare the fractions
PST201F/201/1/2019
by making use of a number line. (You should be able to arrange
the fractions from small to big.) Make sure that you choose the whole correctly, and make
accurate drawings.
8. Use a standard algorithm to calculate: 5 
1 1

4 3
5 1 1
 
1 4 3
5 1 1
 
1 4 3

60  3  4
12

59
12
4
LCD
= 5  12  1  3  1  4
OR
1 12

4 3
3 4
60 3 4
 
12 12 12
11

12
4
59
12
11
12
1 1
9. Use a number line to illustrate to a Grade 5 learners how to add 4  .
3 4
1 1 13 4 1 3
4     
3 4 3 4 4 3

52 3
55


12
12 12
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10. A learner in your class has a problem with mixed numbers. Identify two misconceptions that
3 24
learners might have in solving the problem 8 
and explain how you can prevent such
5 8
misconceptions.
Possible Misconceptions:

Treat numerator and denominator as separate whole numbers and add numerators
together
(3 + 24 = 27) and denominators together (5 + 8 = 13).

Fail to convert mix fraction to improper fraction and find common denominator (LCD).
Preventing Misconceptions:

Mixed numbers must be converted to improper fraction.

Common denominator, LCD of 5 and 8 must be found.
11. You have
3
1
1
kilograms of maize meal. It takes kilogram of maize meal to prepare a full serving
4
2
for your guests. How many full servings can you make?
Maize available = 3
1 7
 kg convert to improper fraction
2 2
One full serving needs
Total servings:
1
kg
4
7 1 7 4
  
2 4 2 1
 14
or
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1
serve 1 person then 1 whole will serve 4 people so we have 3 wholes and half so
4
4  3  12 then 2 quarters makes 1 half so 12+2 =14
If
OR
Therefore there will be 14 full servings for my guests
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PART E
No.
Describe in your own words the first three Van Hiele levels of geometric thought (levels 0, 1 and
1
2). How do the activities, which you will give learners on the three levels, differ?
LEVEL 0: VISUALIZATION/BASIC VISUALISATION/ RECOGNITION
Learners recognize figures by their shape as a “whole” and compare the figures with their
everyday things. Learners recognize how a figure looks like. They use simple language. Learners
do not identify the properties of geometric figures.
LEVEL 1: ANALYSIS/ DESCRIPTION
Learners start analyzing and naming properties of geometric figures. They do not see
relationships between properties. They do not see a need for proof of facts discovered.
LEVEL 2: ABSTRACT/INFORMAL DEDUCTION/ RELATIONAL
Learners are able to make a connection or relationship between properties and figures. They
create meaningful definitions. They are able to give simple arguments to justify their
reasoning. They can identify the minimum requirements for a shape. For example, they know that
a square is a special kind of a rectangle, because they understand that all the other properties of a
square are included in the definition
2
Describe the difference between a polygon and a non-polygon. Use illustrations to support your
answer.
A polygon: A two-dimensional closed figure made up of line segments joined end to end.
A non- polygon: Figures that are not formed by segments (curved), figures that are not closed
and figures with sides that crosses in the middle.
3
3.1
Isosceles, obtuse-angled triangle
3.2
Rhombus
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4
4.1
Regular triangular pyramid: Tetrahedron
4.2
A hexahedron that is a pyramid: pentagonal pyramid
5
Which quadrilaterals are described by the following characteristics? (Name them.) Make a neat
drawing of each. (Do not assume properties that are not given.)
5.1 Square
5.2 Kite
6
7
Is a rectangle a special kind of square? Motivate your answer.
No. It is not a square because all sides of a square are equal, and a rectangle has only opposite
sides equal.
Complete the following table to classify and describe the 3D objects. Name each of the objects (a
mathematical name):
3D objects
a
Mathematical
No. of
No. of
No. of
Polyhedron or
name
faces
edges
vertices
Non-Polyhedron
Rectangular
based prism
6
12
8
Polyhedron
18
d
Sphere
0
0
0
Non-polyhedron
e
Pentagonal
7
15
10
Polyhedron
prism
8.1
Cylinder
8.2
Pentagonal prism
9.1
Octahedron
9.2
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Octagonal pyramid
19
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The front, top, and side views of the given structure are as follows:
Front view
Top View
Side View
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The structure of the 3-D object will be as follows:
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EXAM GUIDELINES MAY/JUNE 2019
NB: A College decision has been made that lecturers are not to demarcate scope specific work for examination
purposes, but that examination questions should be based on all the work covering the notional hours of the
modules. Students should study everything. Where other competencies or skills are assessed differently
during the tuition period, the lecturer in Tutorial Letter 201 will spell out the various assessment methods clearly.
It is important also to note the following information:
- NO CALCULATORS WILL BE ALLOWED.
- Students must bring drawing equipment (pencil, ruler, sharpener, eraser, protractor and compass).
- Your drawings must be neat and eligible. No sketchy drawings will be accepted
- The question paper consists of five (5) questions.
- Work through the solutions of the assignments (in TUT letter 201) carefully. They serve as a guide on how to
approach -- examination questions.
NB: The exam paper covers ALL the work for this module
Theories on teaching and learning
 Piaget’s theories
 Rote learning
 Problem solving
 Mathematics Learning
 Relational and instrumental understanding
 Assimilation, accommodation, disequilibrium
and reflective thought
 Doing and Understanding Mathematics
 Strands of mathematical proficiency
 Cognitive schemes
 Learning theories
 Lesson plans
 Higher/lower order tasks
 Conceptual and procedural knowledge
Content for the module
 Place value
 Whole numbers
 Dienes Blocks
 Number lines
 Comparison of fractions
 Standard algorithms
 Models to illustrate mathematics concepts
 Models to illustrate operations on numbers
 Rules of divisibility
 Compensation method
 Prime numbers
 Models to illustrate fraction concepts
 Spatial sense
 Van Hiele and levels of geometrical thinking
 Properties of polyhedra
 Perspective drawing
 2D shapes and 3D Objects
 Nets of 3D objects
Best wishes
Mrs SM Kodisang
kodissm@unisa.ac.za