the greatest common divisor and the least common multiple of two numbers
A. Basic information of the topic
Example 5: Angle (degree)
3. Teachers may introduce the unit of angle as “degree” and it is expressed by the symbol “ ° ”.
Teachers may use a protractor to illustrate the concept: assume that the circumference of a circle can be divided into 360 equal parts, two adjacent radii will form an angle of 1 degree, and each of the angles formed at the centre of the circle is 1 degree, denoted “1° ”.
4. Teachers may describe the angles at centre of 1/3 circle, 1/4 circle and 1/6 circle by using “degree”
to strengthen students’ understanding of the unit.
5. Teachers may further guide students to describe acute angles, right angles and obtuse angles in degrees, and then introduce the names and concepts of straight angles, reflex angles and round angles.
acute angle right angle obtuse angle straight angle reflex angle round angle
Greater than 0º
and less than 90º 90º Greater than 90º
and less than 180º 180º Greater than 180º
and less than 360º 360º
6. Teacher may introduce the protractor and its use to students. Students are required to recognise the centre, the edge and the degree mark of the protractor.
Steps to measure an angle from 0° to 180° with a protractor:
(1) Place the centre of the protractor to the vertex of the angle;
(2) Align one arm of the angle with the edge of the protractor and measure the size of the angle using the set of degree mark which the arm on the edge of the angle points to “0”.
degree mark edge centre degree mark
(3) Read the degree mark where the other arm is pointing. The reading indicates the size of the angle in degree. For example, ∠BAC in the previous illustration is 38°.
[Note: Students may select the correct degree from the two sets of degree mark by recognising whether the measured angle is an acute or obtuse angle and hence inferring the range of degrees of the measured angle.
7. Teachers may conduct Activity 1 (see the next page) with students to measure the reflex angle with a semi-circular protractor.
8. Teachers may illustrate how to draw an angle of required size (from 0° to 180°) with a protractor:
(1) Draw a line segment (i.e. an arm of the angle).
(2) Place the edge of the protractor along the line segment and place the centre of the protractor at the vertex (i.e. one end of the arm).
(3) Find the required mark using the set of degree mark of which the line segment is pointing
“0”, and then mark a small dot at the circumference of the protractor at the required mark.
(4) Join the small dot to the vertex with a ruler to form the second arm of the angle.
(5) Label the angle with capital letters and mark the degree on the figure.
9. Teachers may conduct Activity 2 (see the next page) with students to draw the reflex angle with a semi-circular protractor.
A
B
C
Activity 1
Measure the size of the angles shown in the following questions with a protractor, write down the answer in the horizontal lines, and mark the degrees on the figures.
1.
∠ABC = ______________
2. (a) Measure ∠ABC
∠ABC = ______________
(b) reflex angle ABC = __________
3.
reflex angle PQR = __________
4.
reflex angle ABC = __________
Activity 2
Draw the required angles shown in the following questions with a protractor, and mark the degrees on the figures.
1. ∠ABC = 130º 2. (a) ∠PQR = 145º
(b) reflex angle PQR = 215º
3. reflex angle ABC = 250º 4. reflex angle PQR = 300º A
B C
A
B C
Q
P
R A
B C
C. Exercise
Measure the size of the angles with a protractor, mark the degrees on the figures, and write down the sizes and types of the angles.
Figure Size of angle Type of angle
1.
∠AOB = ______
acute angle / right angle / obtuse angle / straight angle / reflex angle / round angle
2.
∠ABC = ______
∠BAC = ______
∠ACB = ______
___________
___________
___________
3. Measure the sizes of angles of the triangle (label the vertices with your own choices of letters)
______________
______________
______________
___________
___________
___________
4. Measure the sizes of angles of the triangle (label the vertices with your own choices of letters)
Arranged by the sizes of angles from small to large
______________
______________
______________
______________
______________
______________
5. Draw an angle of the given size.
∠PQR = 210° _____________
A
B O
A
B C
Suggested answer:
Activity 1
Measure the size of the angles shown in the following questions with a protractor, write down the answer in the horizontal lines, and mark the degrees on the figures.
1.
∠ABC = _____ 110º _____
2. (a) Measure ∠ABC
∠ABC = _____ 80º _____
(b) reflex angle ABC = ___ 280º ___
3.
reflex angle PQR = _____ 240º _____
4.
reflex angle ABC = _____ 320º _____
Activity 2
Draw the required angles shown in the following questions with a protractor, and mark the degrees on the figures.
1. ∠ABC = 130º 2. (a) ∠PQR = 145º
(b) reflex angle PQR = 215º
3. reflex angle ABC = 250º 4. reflex angle PQR = 300º
80º
240º
250º
130º
300º 110º
A
B C
320º
215º A
B C
Q
P
R A
B C
A
B C
B
A
C
145º Q
P
R
Q
P
R
Exercise
Measure the size of the angles with a protractor, mark the degrees on the figures, and write down the sizes and types of the angles.
Figure Size of angle Type of angle
1.
∠AOB = 30°
acute angle / right angle / obtuse angle / straight angle / reflex angle / round angle 2.
∠ABC = 90°
∠BAC = 53°
∠ACB = 37°
right angle acute angle acute angle 3. Measure the sizes of angles of the triangle (label
the vertices with your own choices of letters)
∠ABC = 50°
∠BCA = 60°
∠CAB = 70°
acute angle acute angle acute angle
4. Measure the sizes of angles of the triangle (label
the vertices with your own choices of letters) Arranged by the sizes of angles from small to large
∠ABC = 120°
∠BCA = 20°
∠CAB = 40°
obtuse angle acute angle acute angle 5. Draw an angle of the given size.
∠PQR = 210° reflex angle A
B O
A
B C
A
B C
30°
A
C B
210°
P
Q R
90° 37°
53°
70°
60° 50°
40°
20° 120°
Example 6: Pie chart
A. Basic information of the topic
Strand: Data Handling
Learning objective: 1. Recognise pie charts.
2. Interpret pie charts.
- Students are not required to measure the angles at centre of a pie chart for calculations.
- Teachers may let students use IT to construct pie charts.
Prerequisite knowledge: Students should have learned angle (degree) through the learning and teaching materials for the transitional period of the revised junior secondary Mathematics curriculum.
Curriculum planning: Teachers may introduce this topic in Learning Unit 29 “Presentation of data”, or other learning units whichever appropriate.
[Note 1: The learning objective of pie charts in the revised primary Mathematics curriculum only requires students to interpret pie charts involving simple calculations, such as the case that the angle at centre of each sector is a multiple of 30º or 45º. In KS3, students should be able to handle more complex situations. Therefore, this learning and teaching materials does not follow the above restrictions, but teachers may start with simpler pie charts in teaching.]
[Note 2: In this topic, students understand the concept of sectors and angles at centre intuitively. Therefore, this learning and teaching material may not necessarily be arranged after Learning Objective 16.2.]
B. Suggested teaching content
1. Teachers may introduce that pie chart is one of the commonly used statistical charts to express the proportion of each of the item to the whole data set. The shape of a pie chat is circular and each sector represents a corresponding item. The name of items or its percentage are often indicated in or near the corresponding sectors (as shown in the figure below *).
*
We may also indicate the angle at centre of the sectors instead of the percentage in teaching. The percentage of each item of the whole data set in pie chart is directly proportional to the angle at centre of the sector representing the item. For example, for 25% of the whole data set, the angle at centre of sector is 25% of a round angle (i.e. 360º × 25% = 90º). The following formula can be used to find the angle at centre of the sector representing the item:
angle at centre = 360º × percentage of the item = 360º × Frequency of the item Total frequency
2. Teachers may let students use IT to construct pie charts. Teachers may also introduce the method to construct a simple pie chart with paper and pencil:
Step (1): Calculate the percentage of each item and the angle at centre of the corresponding sector.
Step (2): Draw a circle and divide the circle according to the angle at centre of each sector.
Step (3): Mark the sectors by the corresponding items and their percentages (or angles at centre).
Step (4): Write down the title of pie chart.
3. Teachers may conduct Activity 1 (see the next pages) with students to construct a pie chart to show the allocation of funds for the Christmas event and discuss the pie chart with classmates.
4. Teachers may conduct Activity 2 (see the next pages) with students to discuss the application of pie charts in daily lives. Teachers may further illustrate on pie charts, and discuss the pros and cons of using the pie chart with the students. For example, it is easy to find the percentage of different items by the size of sectors intuitively. However, when there are too many items, it is difficult to display the proportion of different items.
Activity 1
1. Suppose your class will spend $720 of the class fee as a fund for the Christmas party. How do you think should the fund be used? Describe your suggestions in five items (including food, drinks, prizes, decoration, and other miscellaneous items) and fill in the form below.
Item Amount Percentage
(corr. to 1 decimal place)
Angle at centre (corr. to the nearest integer) 1 Food
2 Drinks 3 Prizes 4 Decoration
5 other miscellaneous items
Total amount: $720 100% 360º
2. According to the table, construct a pie chart to display your opinion with the aid of a protractor.
3. Exchange your pie chart with your classmates, and discuss which items should be the major spending of the fund.
Activity 2
The pie chart below is the 2018–19 total government expenditure in the 2018–19 Budget.
1. Find the value of x.
2. If the expenditure on education is $110 billion, what is the total government expenditure in 2018 – 19? What is the expenditure on social welfare?
Solution:
Economic 4% Environment and Food 5%
Security 10%
Health 14%
Infrastructure 15%
Others 16%
Social Welfare x % Education
20%
2018-19 Total Government Expenditure
Economic 4% Environment and Food 5%
Security 10%
Health 14%
Infrastructur e 15%
Others 16%
Social Welfare x % Education
20%
2 0 1 8 -1 9 T o t a l G o v e r n me n t E x p e n d i t u r e
Suggested answers:
Activity 1 (open-ended question)
1. Suppose your class will spend $720 of the class fee as a fund for the Christmas party. How do you think should the fund be used? Describe your suggestions in five items (including food, drinks, prizes, decoration, and other miscellaneous items) and fill in the form below.
Item Amount Percentage
(corr. to 1 decimal place)
Angle at centre (corr. to the nearest integer) 1 Food (for example) $360 $360
$720 × 100% = 50% 360 º× 50% = 180º 2 Drinks
3 Prizes 4 Decoration
5 other miscellaneous items
Total amount: $720 100% 360º
Activity 2 1. Since the total percentage of all items is 100 %.
20% + 4% + 5% + 10% + 14% + 15% + 16% + x % = 100%
x % + 84% = 100%
x % = 16%
x = 16 2. Let the total government expenditure for 2018–19 is
$ y billion.
y × 20% = 110 y = 550
∴ The total government expenditure for 2018–19 is
$550 billion.
Social welfare expenditure:
550 × 16%
= 88
∴ The social welfare expenditure of the government in 2018–19 is $88 billion.
Points to note:
1. The learning and teaching material is mainly for the construction and interpretation of pie charts.
Choosing appropriate statistical charts to present data is the learning objective in Learning Unit 29 “Presentation of data” of the revised junior secondary Mathematics curriculum.
2. The teacher may explain to the students that when using a protractor to construct pie chart, the degree of accuracy of angles at centre is usually correct to the nearest integer only. It is likely to take an approximation in calculating the angles at centre, and the sum of the approximated angles at centre may not be 360º. Teachers may point out that there is hence an advantage of using information technology to construct a pie chart concerning the degree of accuracy in presenting data.
C. Exercise
1. Based on the information of total government revenue in the 2018–19 Budget, complete the following table:
Item Amount Percentage
(corr. to 1 decimal place)
Angle at centre (corr. to the nearest integer)
1 Profits Tax 160
2 Salaries Tax 50
3 Stamp Duties 100
4 Investment
Income 40
5 Land Premium 120
6 Other Revenue 130
Total amount: $600 (billion) 100% 360º
2. According to the table, use IT or a protractor to construct a pie chart of the total government revenue in the 2018–19 Budget.
Suggested answers:
1. Based on the information of total government revenue in the 2018–19 Budget, complete the following table:
Item Amount Percentage
(corr. to 1 decimal place)
Angle at centre (corr. to the nearest integer)
1 Profits Tax 160 1600
6000 × 100% = 26.7% 360º× 26.7% = 96º
2 Salaries Tax 50 500
6000 × 100% = 8.3% 360º× 8.3% = 30º
3 Stamp Duties 100 1000
6000 × 100% = 16.7% 360º× 16.7% = 60º 4 Investment
Income 40 400
6000 × 100% = 6.7% 360º× 6.7% = 24º
5 Land Premium 120 1200
6000 × 100% = 20% 360º× 20% = 72º
6 Other Revenue 130 1300
6000 × 100% = 21.7% 360º× 21.7% = 78º
Total amount: $600 (billion) 100% 360º
2. According to the table, use IT or a protractor to construct a pie chart of the total government revenue in the 2018–19 Budget.
Reference materials: The 2018–19 Budget https://www.budget.gov.hk/2018/eng/index.html
Profits Tax 27%
Salaries Tax 8%
Stamp Duties Investment Income 17%
6%
Land Premium 20%
Other Revenue 22%
2 0 1 8 -1 9 T o t a l G o ve r nme n t R e v e nue