Physical Address
304 North Cardinal St.
Dorchester Center, MA 02124
UNIT 7: NUMBER
7.1 PLACE VALUE & MULTI-DIGIT NUMBERS
Meaning of Place Value
Place value is the value of a digit based on its position in a number.
Example:
4 3 2 5
Thousands Hundreds Tens Ones
In 4,325
-
4 = 4,000
-
3 = 300
-
2 = 20
-
5 = 5
Teaching Explanation
Each digit “stands” on a place. Moving left increases value ×10.
Place Value Chart
Ten Thousands | Thousands | Hundreds | Tens | Ones
Expanded Form
Writing numbers as the sum of their place values.
Example:
5,462 = 5,000 + 400 + 60 + 2
7 Worked Examples
Example 1
Write place value of 6 in 3,684.
Solution:
6 is in hundreds place
= 600
Example 2
Expand 8,205.
= 8,000 + 200 + 0 + 5
Example 3
Write 9 thousands, 4 tens, 3 ones.
= 9,043
Example 4
Which is greater: 4,562 or 4,652?
Compare hundreds:
5 < 6
So 4,652 is greater.
Example 5
Round 3,478 to nearest hundred.
= 3,500
Example 6
Value of 7 in 27,451?
Thousands place
= 7,000
Example 7
Arrange in ascending order:
3,456; 3,465; 3,546
= 3,456 < 3,465 < 3,546
GTLE Tip
Expect:
✔ expanded form
✔ rounding
✔ ordering numbers
✔ place value identification
7.2 FACTORS, MULTIPLES & PRIME NUMBERS
Factors
A factor divides a number exactly.
Example:
Factors of 12 = 1, 2, 3, 4, 6, 12
Multiples
Multiples are results of multiplying.
Multiples of 4:
4, 8, 12, 16, 20…
Prime Numbers
Numbers with only two factors: 1 and itself.
Examples:
2, 3, 5, 7, 11
Composite Numbers
More than two factors.
Sieve Diagram (Prime Numbers)
1 2 3 4 5
6 7 8 9 10
Cross out multiples.
7 Worked Examples
Example 1
List factors of 18.
1, 2, 3, 6, 9, 18
Example 2
Is 17 prime?
Factors: 1 and 17
Yes → Prime
Example 3
First five multiples of 6.
6, 12, 18, 24, 30
Example 4
Find HCF of 12 and 18.
12: 1,2,3,4,6,12
18: 1,2,3,6,9,18
HCF = 6
Example 5
LCM of 4 and 6.
4 → 4,8,12
6 → 6,12
LCM = 12
Example 6
Is 21 prime?
Factors: 1,3,7,21
No → composite
Example 7
Write primes between 10 and 20.
11, 13, 17, 19
7.3 INTEGERS
Integers include:
Positive numbers
Negative numbers
Zero
Number line:
-3 -2 -1 0 1 2 3
Rules
Same signs → add
Different signs → subtract
Negative × Negative = Positive
7 Worked Examples
Example 1
5 + (-3) = 2
Example 2
-4 + (-6) = -10
Example 3
7 – 10 = -3
Example 4
-3 × 4 = -12
Example 5
-5 × -2 = 10
Example 6
Arrange: -2, 4, -6, 1
= -6, -2, 1, 4
Example 7
Temperature rises from -3°C to 5°C.
Change = 8°C
7.4 FRACTIONS, DECIMALS & PERCENTAGES
Fraction
Part of a whole.
3
----
4
Decimal
0.75
Percentage
75%
Conversion
Fraction → Decimal → Percent
7 Worked Examples
Example 1
½ = 0.5 = 50%
Example 2
0.25 to fraction
= 25/100 = 1/4
Example 3
30% of 200
= 30/100 × 200 = 60
Example 4
Add: ½ + ¼
= 2/4 + 1/4 = 3/4
Example 5
0.6 as percentage
= 60%
Example 6
¾ of 80
= 60
Example 7
Convert 45% to fraction
45/100 = 9/20
7.5 RATIOS, PROPORTION & RATES
Ratio
Comparison of quantities.
Example:
2:3
Proportion
Equality of ratios.
2/3 = 4/6
Rate
Comparison with different units.
km/hr
Diagram
Boys : Girls
2 : 3
7 Worked Examples
Example 1
Simplify 6:12
= 1:2
Example 2
Share GH₵60 in ratio 2:1.
Total = 3
Each = 20
= 40 and 20
Example 3
If 3 books cost GH₵15, cost of 5?
1 book = 5
5 books = 25
Example 4
Solve: 4/5 = x/20
x = 16
Example 5
Speed = 120km in 2hrs
= 60 km/hr
Example 6
Ratio of 30cm to 2m.
Convert 2m = 200cm
30:200 = 3:20
Example 7
Mix juice in ratio 1:4 using 10L.
Total parts = 5
Each = 2L
Juice = 2L
Water = 8L
Great — let’s continue.
Below is UNIT 8: ALGEBRA presented at teacher-training level with:
✅ Full explanations
✅ Classroom teaching notes
✅ Diagrams
✅ 7 solved examples per topic
✅ GTLE exam focus
Take this as professional College of Education material.
8.1 NUMBER PATTERNS
Meaning
A number pattern is a sequence of numbers that follows a rule.
Example:
2, 4, 6, 8, …
Rule: Add 2
Types of Patterns
-
Increasing
-
Decreasing
-
Repeating
-
Growing
Teaching Note
Ask pupils:
“What is changing?”
“Is it addition, subtraction, multiplication or division?”
Pattern Diagram
2 → 4 → 6 → 8
+2 +2 +2
7 Worked Examples
Example 1
Find next number: 5, 10, 15, __
Rule: +5
Answer = 20
Example 2
3, 6, 12, 24, __
×2
Answer = 48
Example 3
20, 18, 16, __
-2
Answer = 14
Example 4
1, 4, 9, 16, __
Square numbers
Answer = 25
Example 5
2, 5, 10, 17, __
Rule: +3, +5, +7
Next = +9
Answer = 26
Example 6
Find rule: 100, 90, 80
Subtract 10
Example 7
Predict 6th term of 3, 6, 9…
Terms:
3,6,9,12,15,18
6th term = 18
GTLE Tip
Patterns appear as:
✔ missing numbers
✔ rule identification
✔ word problems
8.2 ALGEBRAIC EXPRESSIONS
Meaning
An algebraic expression combines numbers, letters and operations.
Example:
3x + 5
x represents an unknown number.
Parts
Coefficient → 3
Variable → x
Constant → 5
Diagram
3x + 5
↑ ↑
Coefficient Constant
7 Worked Examples
Example 1
Write expression: “5 more than a number n”
n + 5
Example 2
Simplify: 2a + 3a
= 5a
Example 3
Evaluate 4x + 2 when x = 3
4(3)+2 = 14
Example 4
Write: “twice a number minus 4”
2x – 4
Example 5
Simplify: 7y – 2y
= 5y
Example 6
Evaluate: 5p when p = 6
= 30
Example 7
Identify variable in 6k + 9
k
8.3 ONE–STEP EQUATIONS
Meaning
An equation shows equality.
Example:
x + 4 = 9
Rule
Whatever you do to one side, do to the other.
Balance Diagram
x + 4 = 9
remove 4
x = 5
7 Worked Examples
Example 1
x + 5 = 12
x = 7
Example 2
3x = 18
x = 6
Example 3
x – 4 = 10
x = 14
Example 4
x/5 = 4
x = 20
Example 5
2x + 3 = 11
2x = 8
x = 4
Example 6
7x = 35
x = 5
Example 7
x – 6 = 2
x = 8
8.4 TRANSLATING WORD PROBLEMS
Teaching Strategy
Key words:
| Words | Operation |
|---|---|
| sum | + |
| difference | – |
| product | × |
| quotient | ÷ |
7 Worked Examples
Example 1
Sum of number and 8
x + 8
Example 2
Five times a number
5x
Example 3
A number decreased by 4
x – 4
Example 4
Half of a number
x/2
Example 5
Solve: “A number plus 6 equals 14”
x + 6 = 14
x = 8
Example 6
Three more than twice a number
2x + 3
Example 7
Difference between 20 and a number
20 – x
UNIT 9 – GEOMETRY & MEASUREMENT
9.1 PRISMS
Meaning of a Prism
A prism is a 3–dimensional shape with:
-
Two identical ends (bases)
-
Flat rectangular faces
-
Uniform cross-section
Common Prisms
-
Rectangular prism
-
Triangular prism
Rectangular Prism Diagram
________
/______/|
| | |
| | /
|______|/
Triangular Prism Diagram
/\
/__\____
| |
|______|
Teaching Explanation
Tell learners:
“A prism looks like a box or tent.”
Properties
Rectangular Prism:
-
6 faces
-
12 edges
-
8 vertices
Triangular Prism:
-
5 faces
-
9 edges
-
6 vertices
7 Worked Examples
Example 1
Identify prism in classroom.
Chalk box → rectangular prism
Example 2
How many faces in rectangular prism?
Answer: 6
Example 3
Edges in triangular prism?
Answer: 9
Example 4
Name shape of base of triangular prism.
Triangle
Example 5
Give two real-life examples.
Book, cupboard
Example 6
How many vertices in rectangular prism?
8
Example 7
Is cylinder a prism?
No (curved surface)
9.2 CARDINAL DIRECTIONS
Meaning
Cardinal directions show position on Earth.
Main points:
North (N)
South (S)
East (E)
West (W)
Compass Diagram
N
|
W ----+---- E
|
S
Intermediate Points
NE, NW, SE, SW
Teaching Note
Use school compound to demonstrate.
7 Worked Examples
Example 1
Opposite of East?
West
Example 2
Direction between North and East?
North-East
Example 3
Face sunrise.
East
Example 4
School is west of church. Church is?
East of school
Example 5
Direction between South and West?
South-West
Example 6
Name 4 main directions.
N, S, E, W
Example 7
Draw compass.
(Use diagram above)
9.3 TRANSFORMATIONS (REFLECTION & TRANSLATION)
Reflection
Flipping a shape over a line.
Reflection Diagram
□ | □
Mirror line
Translation
Sliding shape without turning.
Translation Diagram
□ → □
Teaching Explanation
Reflection = mirror
Translation = slide
7 Worked Examples
Example 1
Flip triangle over vertical line.
Reflection
Example 2
Move square right by 3 units.
Translation
Example 3
Does size change?
No
Example 4
Does orientation change in translation?
No
Example 5
Reflection produces mirror image?
Yes
Example 6
Identify transformation: shape slides.
Translation
Example 7
Is rotation studied here?
No (single transformations only)
UNIT 10 – DATA & PROBABILITY
10.1 LINE GRAPHS
Meaning
A line graph shows how data changes over time using points connected by straight lines.
Parts of a Line Graph
-
Title
-
Horizontal axis (x-axis)
-
Vertical axis (y-axis)
-
Scale/interval
-
Plotted points
-
Line joining points
Diagram (Simple)
Temperature
|
10| *
| *
| *
|*
+--------------
Mon Tue Wed
Teaching Note
Tell pupils:
“Line graphs help us see increase or decrease.”
Steps to Draw a Line Graph
-
Write title
-
Draw axes
-
Choose scale
-
Plot points
-
Join with straight lines
7 Worked Examples
Example 1
Draw graph from table:
| Day | Temp |
|---|---|
| Mon | 20 |
| Tue | 22 |
| Wed | 25 |
Plot points and join.
Example 2
Highest temperature?
Wednesday (25)
Example 3
Lowest temperature?
Monday (20)
Example 4
What trend is observed?
Increasing
Example 5
Difference between Tue and Wed?
25 – 22 = 3
Example 6
If Thursday is 27, continue graph.
Plot at 27.
Example 7
Why use line graph?
To show change over time.
10.2 DATA COLLECTION METHODS
Meaning
Data is information collected for study.
Common Methods
-
Observation
-
Interview
-
Questionnaire
-
Experiment
-
Electronic media
Teaching Explanation
Choose method based on question.
7 Worked Examples
Example 1
Best method to count pupils?
Observation
Example 2
Know favourite food?
Questionnaire
Example 3
Weather data?
Observation
Example 4
Test plant growth?
Experiment
Example 5
News information?
Electronic media
Example 6
Ask parents income?
Interview
Example 7
Why choose questionnaire?
Collect many responses quickly.
10.3 PROBABILITY EXPERIMENTS
Meaning
Probability is chance of occurrence.
Formula:
Probability = favourable outcomes / total outcomes
Range
0 ≤ P ≤ 1
Example Diagram (Coin)
H T
Types
Theoretical
Experimental
7 Worked Examples
Example 1
Probability of Head when tossing coin.
1/2
Example 2
Rolling die: probability of 3.
1/6
Example 3
Probability of even number on die.
Even: 2,4,6 = 3
Total = 6
= 3/6 = 1/2
Example 4
Spinner with 4 equal colours. Red?
1/4
Example 5
Bag has 3 red, 2 blue balls. Pick red?
3/5
Example 6
Impossible event example?
Getting 7 on normal die.
Example 7
Certain event?
Sun rising tomorrow.
Great — this is the final unit of MODULE 2.
This part is very important for GTLE because it moves from calculation to teacher professionalism, curriculum understanding, and pedagogy.
Study this carefully.
UNIT 11 – MATHEMATICS CURRICULUM & TEACHER BELIEFS
11.1 DEFINITIONS OF CURRICULUM
Meaning of Curriculum
Curriculum refers to all planned learning experiences provided by the school to achieve educational goals.
Definitions by Scholars
Ralph Tyler
Curriculum is learning experiences planned and guided by the school.
Taba
Curriculum is a plan for learning.
Wheeler
Curriculum is experiences learners have under teacher guidance.
Teaching Explanation
Curriculum is not only subjects — it includes:
✔ content
✔ teaching methods
✔ assessment
✔ activities
Classroom Example
Teaching fractions using oranges is part of curriculum.
7 Worked Examples (Theory Application)
Example 1
Define curriculum.
Planned learning experiences.
Example 2
Is sports part of curriculum?
Yes (co-curricular)
Example 3
Who implements curriculum?
Teacher
Example 4
Who designs curriculum in Ghana?
NaCCA
Example 5
Mention two parts.
Content & assessment
Example 6
Is hidden behaviour curriculum?
Yes (hidden curriculum)
Example 7
Lesson notes belong to curriculum?
Yes (planned instruction)
11.2 PHILOSOPHY OF STANDARD-BASED CURRICULUM
Meaning
Standard-based curriculum focuses on:
✔ competencies
✔ skills
✔ performance standards
Not only memorization.
Features
-
Learner-centered
-
Competency-based
-
Activity oriented
-
Continuous assessment
Comparison
Old Curriculum → Objectives
New Curriculum → Competencies
Diagram
Knowledge → Skills → Values → Competence
7 Worked Examples
Example 1
What is main focus?
Skills
Example 2
Who is at centre?
Learner
Example 3
Mention two features.
Competency & assessment
Example 4
Why activity-based?
Promotes understanding
Example 5
Role of teacher?
Facilitator
Example 6
Example of competency.
Problem solving
Example 7
Assessment type?
Formative
11.3 PROBLEM SOLVING & CREATIVITY
Problem Solving
Ability to analyze and solve unfamiliar tasks.
Creativity
Ability to generate new ideas or methods.
Teaching Strategy
Use:
-
word problems
-
group work
-
real-life situations
7 Worked Examples
Example 1
Pupil uses bottle tops to count.
Creativity
Example 2
Teacher gives real-life money problems.
Problem solving
Example 3
Why group work?
Share ideas
Example 4
Example of problem-solving skill.
Logical thinking
Example 5
Creative math activity.
Building shapes
Example 6
Why creativity important?
Builds confidence
Example 7
GTLE tests this by?
Scenario questions
11.4 ICT INTEGRATION IN MATHEMATICS
Meaning
Using technology to enhance teaching.
Examples
-
calculators
-
computers
-
projectors
-
mobile apps
Benefits
✔ improves interest
✔ visual learning
✔ fast assessment
Teaching Example
Use PowerPoint to teach graphs.
7 Worked Examples
Example 1
ICT tool for graphs?
Computer
Example 2
Benefit of projector?
Visual clarity
Example 3
Mobile phone used for?
Calculator
Example 4
Internet used for?
Learning resources
Example 5
Teacher role?
Guide learners
Example 6
Challenge?
Electricity
