Key Stage: 3
Strand: Measures, Shape and Space
Learning Unit: The Rectangular Coordinate System
Objective: To enhance understanding of the relation between slopes of perpendicular lines with the help of IT tools
Prerequisite Knowledge: (i) The changes to the coordinates of a point after a rotation about the origin through multiples of 90
(ii) Finding the slope of a straight line
(iii) The relation between the slopes of parallel lines Resources Required: (i) Dynamic geometry apps for tablets (e.g. GeoGebra)
(ii) A video clip on the topic “Coordinates of Points rotate about the Origin”
(iii) Tablets computers
Description of the Activity:
Pre-lesson Preparatory Activity
The teacher asks students to watch the video clip “Coordinates of Points rotate about the Origin” and answer the question “What are the coordinates of a point P(s, t) after a rotation of 90 about the origin?” before the lesson.
Notes for Teachers:
Activities 1 and 2 below are designed to guide students to discover and understand the relation between slopes of perpendicular lines in the lessons. Before the activities, the teacher may first give a brief review on the concepts introduced in the pre-lesson activity to see whether students have any questions.
Example 15
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Activity 1
1. Students are asked to use the dynamic geometry app GeoGebra to
(a) construct a straight line L, which passes through the origin O, and the point P(6, 5);
(b) rotate the straight line L through 90 about the origin to form a new straight line L1 ; and
(c) mark the point Q (which is the image of P after a rotation of 90 about the origin) on L1 by rotating P.
2. Students are then asked to
(a) find the slopes of L and L1 by considering the coordinates of P and Q respectively; and
(b) observe the relationship between the slopes of L and L1 and draw a conclusion.
3. Repeat steps (1) and (2) above a few times with different coordinates of P and verify the conclusion drawn in step 2(b).
4. During the construction process, the teacher may prompt students with the following questions:
(a) [in step 1(b)] What is the relation between L and L1? (b) [in step 1(c)] What have you got for the coordinates of Q ? (c) [in step 2(a)] How do the results relate to the coordinates of P ?
(d) [in steps 2(b) and 3] What conclusion can you draw? Can you think of another way to present your conclusion?
5. Let students have sufficient time for discussion and exploration before drawing any conclusion.
Activity 2
1. Teacher may then repeat Activity 1, but this time the line L passes through the points P(5, 0) and Q(1, 6) and ask students to rotate the line about point R(3, 3) and consider
Example 15
the slopes of the two lines.
2. Students are required to have a group discussion on whether the conclusion in Activity 1 step 3 still holds for lines not intersecting at the origin. Students are required to provide a logical explanation on their conclusion.
Notes for Teachers:
1. It is desirable for students to work in small groups.
2. The teacher should allow ample opportunities for students to discuss and draw conclusions instead of giving them straightforward hints.
This example mainly involves the following generic skills:
1. Information Technology Skills
Use dynamic geometry apps and tablet computers to facilitate learning
2. Problem solving Skills
Identify the main focus of the problem by building connection between the prerequisite knowledge of point rotation and the perpendicularity of straight lines.
3. Self-learning Skills
Check the mastery of prerequisite knowledge for the learning of the new topic through the pre-lesson preparatory activity
Example 16
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Example 16 Volume of Frustums
Key Stage: 3
Strand: Measures, Shape and Space Learning Unit: Mensuration
Objective: (i) To let students appreciate the mathematical achievements of ancient China through understanding the methods in finding the volume of frustums in The 9 Chapters on the Mathematical Art (九章算術)
(ii) To enhance students’ understanding of the cultural aspect of mathematics.
Prerequisite Knowledge: (i) Properties of similar triangles
(ii) Method of dissection in finding volumes (iii) Finding the volume of rectangular pyramids Resources Required: Cube blocks
Extended Reading Materials: Chapter 5 “Shanggong” (商功) of The 9 Chapters on the Mathematical Art
Description of the Activity:
1. Students are asked to find the volume of a square-based frustum, with lengths of upper square and lower square 40 cm and 50 cm respectively, and with height 50 cm.
In this activity, students would try to derive the formula of the volume of a frustum of square bases, namely V (a b ab)h
3
1 2 2
, where V, a, band h being the volume, length of upper square, length of lower square and height of the frustum respectively.
2. The teacher introduces the method of dissection (棊驗術) used in the problem of
“Square Pavilion” (方亭) in Chapter 5 of The 9 Chapters on the Mathematical Art:
a. As the basic units in the discussion of the volume of a solid, the solids “li-fang”
( 立 方 ), “qian-du” ( 壍 堵 ), “yang-ma” ( 陽 馬 ) and “bie-nao” ( 鱉 臑 ) are
Example 16
introduced to students, which may be done with the aid of computer software on cutting a li-fang gradually, with an example of illustration as Figures 1a and 1b below;
Figure 1a Figure 1b
b. Students are guided to explore the ratio of the volumes of li-fang, qian-du and yang-ma (or cuboid, right-angled triangular prism and rectangular pyramid with the vertex vertical above one of the vertices of the rectangular base) of the same dimension (i.e. qian-du and yang-ma being cut from the li-fang being considered);
c. Students are guided to explore how a square-based frustum could be vertically dissected into li-fang, qian-du and yang-ma and fill in a table as below, which corresponds to the top view of dissection.
yang-ma qian-du yang-ma
qian-du li-fang qian-du
yang-ma qian-du yang-ma
Figure 2 below shows an example of dissection. Students should note that the dimensions of li-fang, qian-du and yang-ma resulted from the dissection are not necessarily the same (i.e. not being cut from the same cuboid).
Figure 2
d. Students are guided to consider multiplying the number of solids in the table
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with the same number so as to represent the total volume in each cell of the table as integral multiples of some li-fangs, i.e. after multiplying the table by 6;
2 li-fang 3 li-fang 2 li-fang 3 li-fang 6 li-fang 3 li-fang
2 li-fang 3 li-fang 2 li-fang
(li-fangs in the above table may have different dimensions)
e. Students are given some cube blocks to visualise the table as a model (as in Figure 3 below), and hence guided to explore how the formula of the volume of a frustum in Step 1 can be derived through the model.
Figure 3
Remark: Students should be reminded that the dimensions of the solids represented by each cell in the tables of (c) and (d) above may NOT be the same, and that the term “li-fang” does not necessarily mean a cube as the modern terminology does. The cube blocks should only be understood as a tool to simulate the abstract ideas of the method introduced.
3. (Enrichment) Students discuss the generalisation of the formula of volume to general prismatoids (i.e. a polyhedron with all vertices lying on two parallel planes) of square or rectangular bases (such as wedges and truncated wedges) by using a similar method as in Step 2 on frustums of rectangular bases. Students are expected to appreciate that the method stated in The 9 Chapters on the Mathematical Art is indeed a more generalised and powerful method to find the volume of prismatoids as compared with applying the properties of similar triangles.
Example 16
Notes for teachers:
1. The key feature of this method of dissection builds on the use of the basic units, known as qi (棊) in the original text. In this example, the basic units li-fang, qian-du and yang-ma are used. It should be emphasised that the ratio of the volumes of these basic units is fixed if they are of the same dimensions.
2. It should be noted that when comparing the basic units of two different cells in the table in Activity 2c or 2d, they may not be of the same dimensions, which means that there is no assumptions that the frustum is made from a right pyramid. Hence, instead of finding the volumes of each solid after dissection, the multiplication done in Activity 2d is essential to form the parts of solids of which the volumes can be found using the given dimensions in the question.
3. After a suitable multiplication to represent each cell of the table in li-fang, there is no need to know all the dimensions of each basic unit. Only the lengths and widths of the upper and lower bases together with the height of the prismatoids are sufficient for finding the volume. In other words, students could be guided to understand that the information about relative positions of the upper and lower bases is not necessary to find the volume of this of kind prismatoids. To students with higher ability and more interest in mathematics, this can serve as an entry point to the important concept of Cavallieri’s Principle.
4. Teacher could use the 3-D printing technology to make a model for illustrating the dissection of the rectangular based frustum (as in Figure 3a and 3b below). This allows students to understand the concept of the method of dissection through hands-on manipulation of the model.
Figure 3a Figure 3b
5. Though the original Chinese terminologies used in The Nine Chapters on the Mathematical Art may arouse the interests of some students, the use of these terminologies is not essential if it would hinder students’ understanding of the mathematical concepts. However, students more capable of understanding classical Chinese texts should be encouraged to read the original remarks by LIU Hui (劉徽)
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to have a taste of the mathematical quality of ancient Chinese mathematicians.
Samples of images of the book can be found in some free electronic library (Figure 4).
Figure 4
This example may serve the purpose of promoting students’ understanding, appreciation and interest in mathematics in the cultural and historical aspects.
This example mainly involves the following generic skills:
1. Creativity
Appreciate and elaborate an alternative methods in finding the volume of a frustum
2. Communication skills
Understand and analyse the classical Chinese text and translate them into modern mathematical language and illustration
3. Problem Solving skills
Deriving the formula of volumes of frustums by referring to solids with known formulae of volume, through planned dissections and multiplications.
Example 17