Similar Triangles and Heights of Objects Brief Lesson Plan

Transcription

1 Similar Triangles and Heights of Objects Brief Lesson Plan Class: Algebra I or II or Geometr Our curriculum doesn t have a separate Geometr course and therefore Algebra II isn t as difficult as some, so I will be doing this activit in Algebra II. But I will wait until we have thoroughl discussed similar triangles and proportions. The students should have prior knowledge of congruent or similar triangle in almost a proof tpe format. There is an option for using trigonometr with a clinometer and for writing the results of Activit 3 in Geometer s Sketchpad. Overview of two da Activit: As ou ma recall, Connections and Applications of mathematics is one of the new NCTM Standards. These activities in this unit relate directl to this Standard. 1) Similar Triangle; Historical Importance. Shows the historical relationship of the ancient Greeks calculating heights of pramids. ) Similar Triangles; Dilations and Architecture. Shows a real life use of similar triangles. 3) Similar Triangles; At home and at the Beach. This is a homework sheet for students to practice with computing proportions. 4) There are three activities: 1. Activit 1 Calculates the height of plaground equipment using mirrors.. Activit Calculates the height of the school gmnasium, using angles of vision, ee height, and ou could use some trigonometr with a clinometer. 3. Activit 3 Students work this one at home. The assignment requires them to find a tall object and use proportions and shows to find the height.

2 Similar Triangles and Heights of Objects Brief Lesson Plan DAY 1 Class: Algebra II Material: Worksheets: 1) Similar Triangle; Historical Importance. ) Similar Triangles; Dilations and Architecture. 3) Similar Triangles; At Home and at the Beach. homework 4) Activities to introduce: 1) Plaground Equipment: Activit1 ) Gmnasium: Activit 3) Choose Your Own Project: Activit 3 Groups overhead Procedure: 1. Opening material. (i.e. correct homework, ask questions, etc.). Handout first two worksheets, Historical Importance and Dilations & Architecture. 3. Read through Historical Importance and walk through Thales procedure on the board or overhead. 4. Read through and observe pictures in Dilations and Architecture.. Handout activities 1,, and 3 and discuss the procedures to tomorrow s activities. 6. Thoroughl read through activities so students will be read to come to class tomorrow and work. You might also want to demonstrate the use of the clinometer. 7. Displa the groups overhead and within the groups the students need to measure each of their own ee heights and record. 8. Tell students to bring mirrors and tape measures or ard sticks to school or our school could have these. 9. Handout homework, Similar Triangles; At Home and At The Beach, and do 1 problems together in class. The rest of the worksheet is homework and due tomorrow.

3 Similar Triangles and Heights of Objects Brief Lesson Plan DAY Class: Algebra II Material: Worksheets 1) Plaground Equipment: Activit 1 ) Gmnasium: Activit 3) Choose Your Own Project: Activit 3 4) Grading Rubric Mirrors (1 per group) Yard sticks/tape measures (1 per group) Groups (3 4 per group) Clinometers (optional 1 per group) Procedure: 1. Hand in homework, Similar Triangle; At Home and At The Beach.. Sit students in their groups. Make sure each groups has a mirror and tape measure/ard stick. 3. Be clear to the students that the group members are required to help each individual in their group measure distance and do other calculations, but each students must use their own measurements, ee height, and sight of view to calculate the heights. * Activit 3 Our gm roof is sloped so I chose to let the students choose whether the wanted to find the height of to the peak of the roof or the height of the walls. 4. Instruct the groups that the must get the measurements needed for activities 1 and before class is done. DON T worr about the calculations until later!. This is a teacher preference, but net I would walk the students to the gm and leave half the groups in the gm and half in the plaground. 6. Allow the students the rest of the time to get their measurements as the teacher walks around answering questions. 7. Meet back in the classroom when the are finished to work on the calculations. 8. Give students 3 etra das in case the need to re measure some things. 9. Activit 3 should be completed on their own. The can get help from others in measuring. Allow about a week for this assignment to be completed. (optional: Handout grading rubric so students see how ou are grading this.)

4 Similar Triangles; Historical Importance Name Date Shadow reckoning was one of the great arts of the ancient Greeks, and was used etensivel b earl mathematicians, especiall in measuring the heights of inaccessible objects. When the Greek mathematician, Thales, visited Egpt, he astonished the people with his use of shadow reckoning to find the height of the Great Pramid. He first waited until his shadow was eactl as long as he was tall, then measuring the length of the shadow of the pramid. What was his difficult with this idea? Of course, he couldn t get into the pramid to measure the distance from the center of the tip of the pramid to the tip of the shadow. He could have known the pthagorean theorem, but that was not historicall recorded. There are was of computing this distance using proportions and few adjustments to this unknown length: Discussion: As stated, Thales waited until the length of the shadow was the same as the length of the stick. What is the value of this ratio? The height of the Great pramid was equal to what other length? The beautiful image of the Pramids below came from Nova's fascinating website, devoted to Egptolog and the Pramids.

5 Similar Triangles; Dilations and Architecture Name Class Similar triangles are one of the most useful topics in geometr, in terms of applications to "real life"! Similarit is a concept that is the basis of scale drawing in architecture and engineering, used in building scale models from to model airplanes to scale models in industr and architecture, and is ver useful in measuring the heights of inaccessible objects. This tpe of measurement is useful for finding the heights of buildings and mountains, and even distances in navigation. When introducing the concept of similar triangles, it is useful to revisit "Transformations" which we discussed previousl in the tet. The overhead projector is perfect to demonstrate "dilation" the original triangle on the overhead projector itself, and the projected image is the dilated triangle. In each diagram, the blue triangle is the original triangle, the green triangle is the dilated image (each side is twice as long as the corresponding side of the original triangle), and point P is the center of dilation. The red "ras" show the dilation. Architects use scale drawing to design houses, hotels, and all tpes of projects. The following floor plan is an eample of a simple house design, originall drawn at the scale 1/4" 1' 0". drawing is "similar" to the actual floor plan of the house: ever 1/4" on the drawing would represent 1 foot in the actual house when it is built. The drawing below is reduced to fit on this page. Scale drawings like this are essential in all aspects of architectural design and in the construction business, as well as in man other design fields, from airplane design to the design of everthing from tos to cities.

6 The floor plan above came from a fascinating and comprehensive website on architecture, which is worth eploring beginning at the following link. This website has hundreds of links to websites on architecture: The beautiful perspective drawing below of a home in Mauna Lani, Hawaii. was designed and drawn to scale b an architectural firm called Architects Studio, whose offices are in Santa Barbara, California. You can visit their website at These web pages will teach ou how to do scale drawings: theaterworks.com/highschooltech/howto/paperwk/scale.htm

7 Name Class At Home And At The Beach Similar Triangles; Direction: Recall solving similar triangles. Show all work and find and. If the problem contains decimals, square roots, or fractions, be sure the answer is consistent with the problem. In 1 6, find the unknown labeled values (possibl z). z 1. z j // k 4. j k

8 . 6. B.8 D 8 A Given: AB 3. / / CD E 1.7 C The shadow of a 0 foot light pole projecting down at the same angle of the house at :00 is 7 feet long. Find the height of a house that is 10 feet awa from center of the base of the house to the light pole as illustrated. SHADOW 10 feet

9 8. Answer the following questions pertaining to the picture. a. Are these triangle similar, congruent, or neither? b. How do ou know this? Show work and eplain thoroughl. c. If the tree casts a 4 foot shadow and at the same time our shadow is 7 feet long. Using our height, measure to the nearest inch, find the height of the tree. (You will get a different answer compared to others in class.) Your height is.

10 Name ANSWER KEY Class At Home and At the Beach Similar Triangles; 1. Direction: Recall solving similar triangles. Show all work and find and. If the problem contains decimals, square roots, or fractions, be sure the answer is consistent with the problem. In 1 6, find the unknown labeled values (possibl z). z z z 616. z z z j // k 4. j k

11 . 6. B.8 D 8 A Given: AB 3. / / CD E 1.7 C The shadow of a 0 foot light pole projecting down at the same angle of the house at :00 is 7 feet long. Find the height of a house that is 10 feet awa from center of the base of the house to the light pole as illustrated ft. SHADOW 10 feet

12 10. Answer the following questions pertaining to the picture. d. Are these triangle similar, congruent, or neither? Similar e. How do ou know this? Show work and eplain thoroughl. mlb 90 LA is congruent to LD LC is congruent LF mle 90 Angles of the sun at the same time AA implies AAA LB is congruent of da are congruent. to LE. Triangle ABC is similar to triangle DEF because all angles are congruent. f. If the tree casts a 4 foot shadow and at the same time our shadow is 7 feet long. Using our height, measure to the nearest inch, find the height of the tree. (You will get a different answer compared to others in class.) Your height is ft. or ft. 7 ANSWERS WILL VARY DEPENDING ON HEIGHT.

13 Name Group members Plaground Equipment Activit 1 You are going to be using a mirror to measure the height of plaground equipment. When ou see an image in a mirror, two equal angles are formed (< 1 < ) these angles are called the angles of incidence and reflection. Materials: Mirror Tape measure or ard stick Problems: 1. Work with our group. These steps should be completed b everone in the group separatel, but group members must help everone!. Choose our equipment. 3. Place the mirror on the ground between ou and the plaground equipment. 4. Move back from the mirror until ou can see the top of the object in the mirror.. Measure and record the distance from our heel to the mirror and the base of the object to the mirror. 6. Measure and record our ee height.

14 Using our knowledge of similar right triangles calculate the measurements, then restate the answer in complete sentences. 7. Find the height of the equipment. 8. Find the distance from the top of the object to the mirror.

15 Name ANSWER KEY Group members Plaground Equipment Activit 1 You are going to be using a mirror to measure the height of plaground equipment. When ou see an image in a mirror, two equal angles are formed (< 1 < ) these angles are called the angles of incidence and reflection. Materials: Mirror Tape measure or ard stick Problems: 1. Work with our group. These steps should be completed b everone in the group separatel, but group members must help everone! 3. Choose our equipment. Swing set 3. Place the mirror on the ground between ou and the plaground equipment. 4. Move back from the mirror until ou can see the top of the object in the mirror.. Measure and record the distance from our heal to the mirror and the base of the object to the mirror Measure and record our ee height. 7

16 Using our knowledge of similar right triangles calculate the measurements, then restate the answer in complete sentences. E Q U I P M E N T 8. Find the height of the equipment. (eample is on the swing set) ft. ft ft. ft ft. ft. ft Find the distance from the top of the object to the mirror ft ft , 300, , , 378, ft. ft.

17 Name ANSWER KEY Group members Height of the Gmnasium Activit In this project, ou will be computing the height of the inside wall of the gmnasium. From our sight of view of the corner of the top of the gm walls connected to the basketball similar triangles are formed. This angle of sight if called our angle of elevation. Then using a clinometer and an point of the gm floor ou can use trigonometr to another height to compare with. Gmnasium corner Ee height 7 Y X Height of gmnasium Length 1 6 _ Length _8 (difference of height of basket and our ee height.) Length _1 4 _

18 Materials: Tape measure/ard stick Clinometer Procedure: 1. Go to the gmnasium and align ourself to view the top of the basket or an specific point on the basket that is measurable along the ground and view the corner edge of the top of the gm as illustrated in the diagram. Our gm roof is sloped so choose the peak height or the height of the walls.. Group members, then help measure the lengths needed for each person to record their measurements. 3. Repeat this with everone in our group. Remember that our height will depend upon our placement on the gm floor. Measurements are going to be different, but the height of the gm should be close to the same. 4. (optional) Choose one member of our group to use the clinometer. Again align ourself and find the angle of elevation. All members of the group need to help find alignment and measurements. Record this in the below illustration.. After gathering data, return to class to help each other calculate. Clinometer Reading Angle º Length _1 4

19 Name ANSWER KEY Group members Height of the Gmnasium Activit Answer the following questions and hand in these problems along with the lengths and work recorded on page 1 and. Problems: 1. Did ou measure the height to the peak or top of the walls? (circle). Using our measurements find the height of the gmnasium using proportions Y Y ft ft. b g , 633 ft ft. 40 The height of the gmnasium using proportions is approimatel feet. 3. Using our measurements find the height of the gmnasium using trigonometr. º 1 4 z tan tan z ft. z G I K J F H ft. The height of the gmnasium using trig is approimatel feet.

20 Choose Your Own Object Activit 3 Name Due date 1. Choose a tall object to measure, using indirect measurement. This object must be something that cannot simpl be measured with a ruler or tape measure; it must be something that is ver tall, and inaccessible, such as a tall flagpole, a building, or a light pole. It also needs to be in a relativel flat field, park, or section of town, so that its shadow will be measurable. This ma depend on the time of da that ou do our measuring, so ou will probabl need to make some observations in different parts of town, and at different times of the da. Of course the shadows will be shortest near noon!. Measure the length of our object's shadow, and measure the length of our own shadow as ou stand nearb, or measure the length of the shadow of a ardstick held verticall nearb. If ou use our own shadow, ou ma need an assistant, and ou will need to know our own height. 3. Open a new GSP (Geometer s Sketchpad) file. Draw careful sketches with data accuratel labeled, and find the height of our object, using the method shown in class. (Do not measure segments in GSP use the data from our measurements outdoors!) Write a careful eplanation, including an problems ou encountered. Tell us what our object is, where it is located, at what time ou measurement the shadow length. Show our calculations and label within the picture. Be sure to use accurate values in our calculations, and do not round an numbers until ou get our final answer. Presentation is alwas important; do a nice job on the drawings and the written work, so that it shows pride in our work. Eample: Similar triangles can be ver useful for measuring inaccessible objects. One method of doing this is called "shadow reckoning". To measure the height of a flagpole, for eample, ou would use the following procedure: measure the shadow of the flagpole, then hold a stick verticall nearb, and measure the shadow that the stick casts. As shown in the diagram below, the height of the flagpole is a vertical measurement and the stick is vertical, so we have a pair of right angles. The sun is a fied point in the sk and so ver far awa that we can assume the ras are parallel; the sun's ras create equal angles at the top of the pole and the top of the stick. Therefore the two triangles are similar, b AAA~, and we can write a proportion and find the needed height.

21 Name Group Members Similar Triangles; Heights Grading Rubric DESCRIPTION OF OBJECTIVES AT HOME AND AT THE BEACH Problem Solving Proportions have been set up accuratel 6 parallel lines (3&6) sum of lengths (&7) cross multiplication and computation is accurate pthagorean theorem used correctl eact values are used when necessar rationalized roots Understanding of Triangles thorough eplanation of triangles (8 a & b) 3 unit of measurement is appropriate (labels) 1 PLAYGROUND EQUIPMENT data collection is accurate 4 accurate equation for height and hpotenuse length 3 computations are accurate sentence statement showing understanding HEIGHT OF GYMNASIUM Research data collection is accurate 4 clinometer data is accurate (angle of incline) Problem Solving height with proportions is accurate height with trigonmetr is accurate 4 height addition 1 OWN PROJECT Report organized neat sketch lengths shown in drawing 1 paragraph is thorough 4 includes: object, time of da, problems/research stor Problem solving accurate data collection proportions are accurate solutions are accurate POINTS/ OBJECTIVE STUDENT S GRADE TOTAL POINTS CONVERSION TO TEST POINTS 100

22 Bibliograph Cathleen V. Sanders (1997), I MATH; Similar Triangles, Cathleen V. Sanders Cathleen V. Sanders (1997), Connecting Geometr; Similar Triangles, Cooke, R. (1997). The histor of mathematics: A brief course. New York: John Wile & Sons, Inc. Dunham, W. (1991). Journe through genius. Penguin Books, Inc.