1 Secondary 4 Mathematics Lesson Study Topic: Finding the ratio of the area of triangles Teachers from Shatin Tsung Tsin secondary School: Mr. Cheung Man Wai, Mr. Mak Shing Chit, Mr. Chang Wing Kang, Mr. Fung Kwok Leung, Miss Chung So Sum, Mr Lau Chi Hin, Mr. Choi Wai Man Members from Hong Kong Institute of Education: Dr. Sze Chong Lap, Mr. Nelson Chu, Mr. Micheal Lee This lesson study was carried out in the period of February – May 2004, during which 9 meetings (including 1 post-test conference) were held among six teachers and three research team members in the school. Background Before the lesson study, students should have learnt the ways to find the area ratio of triangles by using similar triangles. Stage 1: Incubation of ideas Choosing and defining the object of learning From teachers’ experiences, students found it difficult to manage the questions related to the area ratio of common height triangles and there is no textbook highlighting this topic. Therefore, our focus was set on finding the area ratio by using the properties of common height triangles and similar triangles. Objectives of this lesson study are as follows. (1) Identify the height of a triangle with respect to given base. (2) Identify the common height of a pair of triangles. (3) Finding similar triangles on a given figure. (4) Finding triangles with common height. (5) Finding area ratio of similar triangles. (6) Finding ratio of sides of similar triangles from area ratio. (7) Finding area ratio of common height triangles. Stage 2: Pretest Setting and Result Analysis A. Design of Pretest A pretest was designed to gather information on students’ prior knowledge and students’ understanding about the ratio of areas of triangles of common height. A pilot test was given to a sample of six S.5 students. Based on their responses, the questions were refined. The pretest was administered to classes 4A and 4B. 2 B. Results of Pretest The result of the pretest are summarized as follows: Qn. Ability testing Percentage of correct answer or method 4A 4B 4A & 4B 1 Identifying the height of a triangle with respect to given base 63.4% 58.3% 61.0% 2 Identifying the common height of a pair of triangles 87.8% 80.6% 84.4% 3a. Finding area ratio of similar triangles 75.6% 55.6% 66.2% 3b. Finding ratio of sides of similar triangles 85.4% 63.9% 75.3% from area ratio. . 3c. Finding area ratio of common height triangles 51.2% 50.0% 50.6% 4a. Finding similar triangles on a given figure 85.4% 88.9% 87.0% 4b Finding triangles with common height 59.8% 72.2% 65.6% 4c Finding area ratio of common height triangles 41.5% 38.9% 40.3% 4d. Finding the ratio of the length by using similar triangle 85.4% 75.0% 80.5% 4e. Finding area ratio of common height triangles 36.6% 13.9% 26.0% 4f. Finding ratio of lengths of similar triangles 92.7% 63.9% 79.2% 4g. Finding area ratio of similar triangles. 56.1% 30.6% 44.2% 5a Finding area ratio of common height triangles 73.2% 58.3% 66.2% 5b Finding triangles with common height 70.7% 63.9% 67.5% 5c Finding area ratio of similar triangles 70.7% 66.7% 68.8% 5d Finding ratio of sides of similar triangles from area ratio. 92.7% 83.3% 88.3% 5e Finding area ratio of common height triangles. 46.3% 41.7% 44.2% 6 9.8% 2.8% 6.5% Problem solving: Finding the area ratio by using the common height and similar triangles. 3 C. Analysis of the results of the pretest 1.) From the results of question 1, it was found that some students drew the height with respect to the horizontal base instead of the line AB as required. 2.) Question 3c was a typical question of finding the area ratio by using the common height of triangles. The results show around 50% students failed to do so. 3.) For question 4c and 4e, only around 40.3% and 26% of students could answer the question correctly respectively. This shows students find it difficult to find the area ratio by identifying the triangles with common height especially when the bases are not horizontal. 4.) For question 4g, the results could reflect some students might forget the relationship between the area ratio and the corresponding sides of similar triangles. 5.) The results of questions 5e and 6 show that students find it even harder to solve the mixed type problems. D. Conclusions drawn from the analysis of the pretest results (V1) 1.) Some students might not know the relationship between the area ratio of common height triangles and their corresponding sides. 2.) Most students could not find the area ratio of common height triangles because they were not able to identify the common height of triangles especially when the base was not horizontal. E. Determination of the critical feature From the results of the pretest, it was found that many students did not know the relationship between the area ratio and the corresponding bases in the common height triangles. From the research of the past CE papers, there are three basic approaches in solving problems related to the area ratio of triangles. They are as follows. 1.) Considering the common angle of triangles and applying the formula of Area 1 ab sin C ; 2 2.) Considering similar triangles and applying A1 l ( 1 )2 ; A2 l2 4 3.) Considering common height triangles and applying A1 b1 . A2 b2 After discussion, the group of teachers determined the critical features were 1) and 2). F. Variation of teaching strategy (V2) This topic is not discussed in common textbooks. The following teaching strategies were discussed. A l 1.) Use HKCEE questions to show the techniques including 1 ( 1 ) 2 and A2 l2 A1 b1 in solving this kind of problems. A2 b2 2.) Explain how the formula Area 1 ab sin C can be used to solve different 2 problems on ratio of areas of triangles. 3.) Ask students to find common height triangles on a figure. Then ask them to draw the common height of the common height triangles. Stage 3: Implementation of the Research Lesson A. Planning of the lesson study One double lesson (80 minutes) would be conducted to go through the above critical features. The planning is as follows. Activity Time required Remarks Height of triangle 5 min Draw the height of the triangle regarding to a specific side as base Revision on the area ratio of similar triangles 10 min Examples 5 min Classwork (Practice) 10 min Check students’ understanding Introduce the concept of finding the area ratio of common height triangle 10 min Introduce and apply the A b formula 1 1 A2 b2 Examples 5 min Classwork (Practice) 10 min Applying A1 l ( 1 )2 A2 l2 Check students’ understanding 5 Activity Time required Remarks Examples on harder problem combined with finding the area ratio of similar and common height triangles 10 min Problem solving technique in solving problems related to the area ratio of similar and common height triangles. Classwork 10 min Check students’ understanding Discussion of the classwork 5 min B. Use of Variation (V3) Learning activity Revise how to find the area ratio of similar triangle Variation Background (Invariant) Critical feature Common height triangles and similar triangles Area ratio of two similar triangles is equal to the square of their corresponding sides. Date Event Class Teacher 25/3 Pilot test 5B LCN 19/3 Pretest 4A/4B CMW/LCN 22/3 Tryout lesson 4D FKL 23/3 Lesson study 4B LCN 25/3 Lesson study 4A CWM 1/4 Post-test 4A/4B CWM/LCN Show how to find the area ratio by identifying the common height triangle and taking the To identify and use the property of common height triangles as the critical Area ratio of common height triangles is equal to feature to solve the area ratio the ratio of their problem. corresponding bases. ratio of their bases. B. Schedule A tryout lesson was given by FKL in 4D. Here are the observations of the tryout lesson: a) Most students could draw the common heights of two triangles. A l b) Most students knew the formula 1 ( 1 ) 2 for similar triangles. A2 l2 c) Most students found no difficulty on finding ratios of sides of similar triangles. d) Some students got confusion between similar triangles and common height triangles. 6 e) Some students did not know how to find the ratio of bases of common height triangles. A l f) Many students wrongly applied 1 ( 1 ) 2 to common height triangles. A2 l2 g) Many students found difficulty on solving mixed type problems. h) There was not enough time to finish the worksheet. After the tryout lesson, we made the following changes to the lesson plan. a) The time for the students to draw the heights of triangles should be shorter. A l b) The time for revision of the formula 1 ( 1 ) 2 should be shorter. A2 l2 c) The difference between similar triangles and common height triangles should be emphasized. A l A b d) The difference and time to apply the formulae 1 ( 1 ) 2 and 1 1 should A2 l2 A2 b2 be emphasized. e) Mixed type problems should be discussed more and in detail. After the 4B lesson, it was suggested to revise the area ratio of similar triangles first as the introduction. Students should then be asked to draw the height of triangles and then draw the common height of triangles. The focus of the lesson would shift to the area ratio of common height triangles. The flow of the lesson could then be more smooth. Also, full size copies of worksheet should be given to students so that there would be enough room for them to write down their steps and answers. After the implementation of the lesson plan on 4A and 4B, we have the following conclusions. a) Students were interested in the recreational problems. b) Most students understood the difference between similar triangles and common height triangles. c) The critical feature was the common height. Students could find the ratio of the areas if they knew where the common height was. Therefore, it was important to ask the students to draw the common heights. d) Students need more training on mixed type problems so that they can break down the problem into smaller part and separate the two concepts themselves. 7 Stage 4: Evaluation A. Result of Post-test The teachers administered the post-test, which was parallel to the pretest, to the students after the research lessons. Since some of the students knew the numerical answers of the pre-test, we changed some numerical values and the way of questioning. The results of the pre-test and post-test are summarized as follows. Qn Ability testing . Percentage of correct answer or method 4A Pre 1 Identifying the height of a triangle 4B Post Pre 4A & 4B Post Pre Post 63.4% 68.3% 58.3% 81.1% 61.0% 74.4% with respect to given base 2 Identifying the common height of a pair of triangles 87.8% 95.1% 80.6% 97.3% 84.4% 96.2% 3a. Finding area ratio of similar triangles 75.6% 90.2% 55.6% 81.1% 66.2% 85.9% 3b. Finding ratio of sides of similar triangles from area ratio. 85.4% 95.1% 63.9% 91.9% 75.3% 93.6% . 3c. Finding area ratio of common height triangles 51.2% 97.6% 50.0% 83.8% 50.6% 91.0% 4a. Finding similar triangles on a given 85.4% 100% 88.9% 91.9% 87.0% 96.2% figure 4b Finding triangles with common height 59.8% 80.5% 72.2% 67.6% 65.6% 74.4% 4c Finding area ratio of common height triangles 41.5% 80.5% 38.9% 67.6% 40.3% 74.4% 4d. Finding the ratio of the length by using similar triangle 85.4% 90.3% 75.0% 81.1% 80.5% 85.9% 4e. Finding area ratio of common height triangles 36.6% 75.6% 13.9% 45.9% 26.0% 61.5% 4f. Finding ratio of lengths of similar 92.7% 90.3% 63.9% 75.7% 79.2% 83.3% triangles 4g. Finding area ratio of similar triangles. 56.1% 63.4% 30.6% 62.2% 44.2% 62.8% 5a Finding area ratio of common height triangles 73.2% 80.5% 58.3% 70.3% 66.2% 75.6% 5b Finding triangles with common height 70.7% 97.6% 63.9% 94.6% 67.5% 96.2% 5c Finding area ratio of similar triangles 70.7% 100% 66.7% 94.6% 68.8% 97.4% 5d Finding ratio of sides of similar triangles from area ratio. 92.7% 97.6% 83.3% 91.9% 88.3% 94.9% 5e Finding area ratio of common height 46.3% 95.1% 41.7% 73.05 44.2% 84.6% triangles. 8 Qn Ability testing . Percentage of correct answer or method 4A Pre 6 Problem solving: Finding the area ratio by using the common height and similar triangles. 4B Post Pre 4A & 4B Post Pre Post 9.8% 56.1% 2.8% 29.7% 6.5% 43.6% 7. Harder Problem solving: Finding the area ratio by using the common height and similar triangles. 41.5% 13.5% 28.2% Here are some observations about the post-test. a) Nearly all students knew how to find the area ratio of similar triangles after the lesson (Q3a, 4g, 5c). b) The number of students knowing how to find the ratio of sides of similar triangles from the area ratio (Q3b, 4d, 4f, 5g) had increased after the lesson. c) The number of students knowing how to find the area ratio of common height triangles (Q3c, 4c, 4e, 5e) had a large increase compared with the pretest. This A b indicated that most students learnt the formula 1 1 for common height A2 b2 triangles. d) Very few students answered Q6 correctly before the lesson and about 43% of them answered it correctly after the lesson. This indicated that many students had difficulty on handling mixed type problem. e) An extra mixed type problem (Q7) was added to the post-test. About 28% of students answer it correctly. The result was satisfactory. f) Cross-tabulation Considering the typical question of area ratio of common height triangle (Q3c and 5e), there were 50.6% and 44.2% of students could get the right answers in the Q3c and Q5e in the pretest respectively. There was a high percentage of students getting the correct answers (91.0% of 3c and 84.6% of 5e) after the research lesson. This showed that student learnt the objectives as we expected. B. Reflection There are several reasons why the topic was chosen. First, many students find this topic very difficult. Second, this topic was not taught formally in previous years. Finally this topic has not been discussed in common textbooks. Most teachers taught this topic by showing some past HKCEE questions and many students reflected that this topic is very difficult. Therefore we chose this topic and hope that the students may learn how to solve this kind of problems. 9 After the lesson studies, we confirm that the critical feature of this topic is the common height. If students can draw the common height of a pair of common height triangles, they can find the ratio of their area. Many students find it difficult because they cannot find the common height. Before the lesson study, many students wrongly A l apply the formula 1 ( 1 ) 2 to common height triangles. After teacher’s A2 l2 emphasizing the difference between the two formulae, the performances of the students have been improved very much. C. Teacher’s professional growth These lesson studies give us an opportunity to share our ideas and methods in teaching and learning in this specific topic. When we discussed on how to teach this topic, we found that there are many different approaches and it is difficult to find which one is the best. After the pretest and the tryout lesson, we are able to find the weakness of students and the critical feature. We found that students could not solve this kind of problems because they could not find the common height. Therefore we emphasized on how to find the common height of a pair height triangles. We also A l A b found that students were confused by the two formulae 1 ( 1 ) 2 and 1 1 . A2 l2 A2 b2 Therefore we also emphasized on the difference between them. It is agreed that this kind of lesson study is helpful and valuable to be continued for both students and teachers if there is sufficient resources. By combining different point of views of teachers, the critical feature can be found much easier. Teachers may also learn from each other. The most valuable output of the lesson study is not the particular lesson plan. By continuous practice of lesson studies and term works, both students and teachers will be benefited. Appendix: 1. Pretest/Post-test 2. Pretest/Post-test with solution 3. Worksheet 4. Coding 5. Sketchpad and Powerpoint files (teaching aids) 10 Shatin Tsung Tsin Secondary School S.4 Mathematics Pre-test Name: ______________________ Class: ______ ( _____ ) Date: _______________ 1. Draw the heights with bases AB for the following triangles. B B A B A A 2. Draw the common heights of ABC and ACD in the following figures. A 1 B C B C 2 D D A 3. a) R In the figure PS // QR , PS 4 and QR 6 . S area of PST : area of RQT T 4 6 5 = _______________________ P Q b) Given that ABC is similar to DEF , area of ABC : area of DEF 1 : 4 . F C A AB : DE B D 6 E c) = ______________________ A area of ABC : area of ACD 4 5 = _______________________ B 2 C 3 D 7 11 4. C 4 D E 6 A 8 B In the figure, it is given that AB 8 , AD 6 and DC 4 . a) Write down all pairs of similar triangles. ______________________________________________________________ ______________________________________________________________ b) Write down all pairs of non-overlapping (不重疊) triangles which have common 3 height. ______________________________________________________________ ______________________________________________________________ c) Find area of ADE : area of DEC . 4 ______________________________________________________________ d) Find CE : EB . 7 ______________________________________________________________ e) Find area of ABE : area of AEC . ______________________________________________________________ f) Find CE : CB . 7 ______________________________________________________________ g) Find area of DEC : area of ABC . ______________________________________________________________ 5 12 5. R S In the figure PS : QR 2 : 3 . T P Q a) Write down all pairs of similar triangles. 3 ______________________________________________________________ b) Write down all pairs of non-overlapping triangles which have common height. 4 ______________________________________________________________ c) area of PST : area of RQT = ________________________________ 5 d) ST : TQ = ____________________________________________________ 6 e) area of PST : area of PTQ = __________________________________ 7 6. D C E In the figure, ABCD is a parallelogram. Given that area of CEF : area of ADE 1 : 4 . F A B Find area of CEF : area of ABCD . ______________________________________________________________ ______________________________________________________________ ______________________________________________________________ 13 Shatin Tsung Tsin Secondary School S.4 Mathematics Pre-test Name: ______________________ Class: ______ ( _____ ) Date: _______________ 3. Draw the heights with bases AB for the following triangles. B B A B A A 4. Draw the common heights of ABC and ACD in the following figures. A 1 B C B C 2 D D A 3. a) R In the figure PS // QR , PS 4 and QR 6 . S area of PST : area of RQT T 4 6 5 = 4:9 P Q b) Given that ABC is similar to DEF , area of ABC : area of DEF 1 : 4 . F C A AB : DE B D 6 E = 1:2 A c) 4 5 area of ABC : area of ACD 7 B 2 C 3 D 14 = 2:3 4. C 4 D E 6 A 8 B In the figure, it is given that AB 8 , AD 6 and DC 4 . h) Write down all pairs of similar triangles. DEC and ABC 3 i) Write down all pairs of non-overlapping (不重疊) triangles which have common height. A D E and DEC A B E and AEC 4 j) Find area of ADE : area of DEC . 3:2 k) Find CE : EB . 7 2:3 l) Find area of ABE : area of AEC . 3:2 m) Find CE : CB . 7 2:5 n) Find area of DEC : area of ABC . 5 15 4:25 5. R S In the figure PS : QR 2 : 3 . T P Q f) Write down all pairs of similar triangles. 3 P S T and RQT g) Write down all pairs of non-overlapping triangles which have common height. P S T and PTQ , PQT and QTR 5 h) area of PST : area of RQT = 4:9 i) ST : TQ = 6 2:3 7 j) area of PST : area of PTQ = 2:3 6. D C E In the figure, ABCD is a parallelogram. Given that area of CEF : area of ADE 1 : 4 . F A 4 B Find area of CEF : area of ABCD . EF : DE 1 : 2 area of CEF : area of CED 1 : 2 area of CEF : area of A C D 1 : 2 4 1 : 6 area of CEF : area of ABCD 1 : 12 16 Worksheet C A C C B B A B A Draw the height if we Draw the height if we regard Draw the height if we regard AB as base. AC as base. regard BC as base. Theorem 1: For any two similar figures, area of ABC l1 area of PQR l 2 2 Example 1. In the figure, BC// DE, AC= 3 cm and CE= 4 cm. Find D B area of ABC area of ADE A 3 cm C 4 cm E Class work 2. In the figure, PSQ, QXR and RYP are straight lines. If the area of PQR is 225 cm 2 , find the P area of the parallelogram SXRY. S Q 20 cm X Y 30 cm R 17 For triangles with common height, Theorem 2: ABD and ADC with common height, area of ABD area of ADC Example 3. Given that BC : DC 5 : 1. A area of ABD area of ADC B C D 4. Given that AD = 3 cm and CD = 1 cm. A area of ABD area of BDC 3 cm D 1 cm C B Class Work 5. (Modified HKCEE 1996) In the figure, find area of PRX : area of QRX . R 5 4 Ans:______________________ P X Q 18 6. In the figure, QX : XR 5 : 6 and P PY : YR 5 : 3 . If the area of PQR is 44 cm 2 , calculate the area of (a) PXR Y Q R X (b) RXY Example 7. (Modified 1999 Q54) In the figure PQRS is a rectangle. M is a midpoint of QR. PR and MS intersect at N. Find area of NRS : PQMN . P S N Q R M Class Work: 8. (Modified 1994 ) In the figure, PX:XQ = 1: 2, PY:YR = 3:2. Find area of QXY : area of PQR . Q X P 9. (Modified 1997 Q53.) In the figure, PQRS is a rectangle. RSX is a straight line and PX// QS. If the area of PQRS is 24 and Y is a point on QR such that QY :YR = 3:1, find the area of SXY . R Y X P Q S Y R 19 10. (Modified 1996 Q35) In the figure, if area of RSX 1 , area of QRX 3 Find S R area of RSX . area of trapezium PQRS X P Q Homework 11. In the figure. PQ// SR. PR and QS intersect at X. Area of PXQ and that of RXS are 16 and 25 P respectively. Find the area of the trapezium PQRS. Q X S R 12. In the figure, PQ// RS. Given the area of PXS is 18 cm 2 and QX: XS =2:3. Find the area of the trapezium PQRS. S P X Q R