(This lesson plan assumes the students are using an air-powered rocket as described in the Materials section.)

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1 The Mah Projecs Journal Page 1 PROJECT MISSION o MArs inroducion Many sae mah sandards and mos curricula involving quadraic equaions require sudens o solve "falling objec" or "projecile" problems, which involve equaions dealing wih graviy. This lesson allows he sudens o experience he relevance of hese equaions by acually launching rockes and calculaing he maximum heigh of heir flighs. The lesson leads he sudens from he experienial o he absrac by using he daa and high ineres conex of he aciviy o analyze he criical componens of quadraic equaions. daa collecion and predicions (This lesson plan assumes he sudens are using an air-powered rocke as described in he Maerials secion.) Before aking he sudens o he launch sie, explain he logisics and safey procedures. Each group of four will have a "somper," "loader," "imer," and "recorder." The loader places he rocke on he launcher. Jus as a safey measure, he loader should never place his/her face over he launcher. The imer should sar he wach once hey see he somper jump on he launch pad, and should sop i once he rocke his he ground. This is done wice by each group and recorded on he suden handou. I is suggesed ha he eacher have wo Somp Rocke Launchers available, and have groups sanding in line for he wo launch saions. Once he sudens are back in class, hey are o 1) describe and skech he physical pah of he rocke, 2) skech he relaionship of he rocke's ime and heigh, which allows he eacher o discuss he difference beween hese wo parabolas, and finally 3) predic he maximum heigh of he rocke (by insinc, no calculaion, ye). Conceps Wriing, graphing and solving quadraic equaions; finding he verex and roos Time: 3 Hours Maerials Air Rockes, which can be made in he classroom and fueled by an air compressor, or can be purchased for approximaely $10 a mos oy sores. The oy versions involve jumping on an air bladder ha propels he rocke upward (Air Power Rockes by Valen Bros Game Co a Toys R US, or Somp Rockes by D&L) Preparaion This lesson is inended o be a summary lesson of quadraics. Sudens should already have a familiariy wih he algorihms involved in solving and graphing quadraics. The eacher will need o choose a venue for he launches. A large area such as he school's fooball field is suggesed. firs launch 4. The sudens should firs ponder he concree meanings of he verex, roos, and y-inercep. a) The ypical somp rocke is 1.5 fee long. Since he wach sops once he ip of he rocke his he ground, hen he iniial heigh is he heigh of he ip of he rocke a launch: 1.5 fee. b) The heigh a he end of he fligh is 0. c) The highes poin will be reached a approximaely half of he oal ime. This is only an esimae since he rocke does no sar a heigh zero. d) The sudens will know all variables excep he iniial velociy. SUBMITTED BY: Dr. Tom Benne, California Sae Universiy San Marcos; David Maoon, Poer Junior High; Ginny Jackson, Chaparral High School; Rober Fizparick & Chris Shore, Grea Oak High School MATH-123

2 The Mah Projecs Journal Page 2 PROJECT Mission o MArs (coninued) 5. By plugging in zero for h() and he recorded ime for, he sudens solve he only unknown variable, iniial velociy, and hen rewrie he equaion in sandard quadraic form. Once his is done, he sudens use he equaion o deermine a able of values and graph his equaion. They should be able o esimae maximum heigh from he graph. second launch This is an assessmen of he sudens' working knowledge of he aciviy. They are asked o use he second fligh ime o conduc all he same calculaions ha hey did wih he firs. This should be done as independenly as possible, wih minimal eacher assisance, in order o check for undersanding. rocke launch analysis Up o his poin he lesson has been focused on concree applicaions of quadraics. This nex and final phase focuses on he absrac meanings of he quadraics componens: verex, roos, y-inercep and soluions. The suden is asked o relae hese componens o he fligh of he rocke as well as he graph. These quesions are imporan for he sudens' concepual undersanding of parabolas and should be srongly emphasized by he eacher hrough class discussion. 7. By using he using he well-known formula x = -b/(2a), sudens can find he coordinaes of he verex, and compare i o heir predicions. 8. By seing he equaion equal o zero, he sudens ge wo values. One of which is he ime ha he rocke his he ground. Bu wha is his oher value, and why is i negaive? I is he oher ime when he rocke was a heigh zero or should we say, would have been a zero. I is negaive, because a ime zero, he rocke was already a a heigh of 1.5 fee. If we were o follow he graph back in ime (hus, he negaive roo), we see when he rocke would have hypoheically launched from heigh zero. 9. The y-inercep can be seen direcly in he equaion and should be 1.5 fee he iniial heigh of he rocke a ime zero. 10. The sudens evaluae for y given x = 2. Have sudens plo his poin. 11. The sudens se he equaion equal o 10 and solve. This gives conex o wha i means o solve for x given y. There are wo soluions because he rocke is en fee off he ground wice! Why is here only one soluion for #10? Have sudens plo boh soluions. 12. The sudens should ge an imaginary soluion. There are no real roos because he rocke really does no go ha high MATH-123

3 The Mah Projecs Journal Page 3 MISSION Have you ever launched a rocke or hrown a ball as hard as you can - sraigh up? Ever wonder how high i wen? Le's find ou! You are o launch a rocke and ime how long he rocke akes o reurn o earh. For he sake of he aciviy, you will need wo launches. Be sure ha you launch wice and record boh imes. Launch Time (seconds) o MArs Firs Trial Second Trial 1. Observe he behavior of he rocke as i is being launched and raveling hrough he air. Describe how he rocke ascends, urns and descends. Also, skech he pah of he rocke below. h 2. Skech wha you hink he graph of he heigh of he rocke (range) versus ime (domain) looks like. d h 3. How high do you hink your rocke wen?

4 The Mah Projecs Journal Page 4 Mission o MArs: Firs LAunch The formula for he rocke fligh is h() = v 0 - ½ g 2 + h 0, where... h() = heigh v 0 = iniial velociy = ime h 0 = iniial heigh g = he force of graviy (32 f/sec 2 ) 4. a) Wha is he heigh a he beginning of he fligh? b) Wha is he heigh a he end of he fligh? c) When will he highes poin of he fligh occur? d) There is a poin during he fligh when we know all he variables, excep one unknown? Wha is ha variable? 5. a) Use your fligh ime o calculae your iniial velociy, v 0. h() = v 0 - ½ g 2 + h 0 Disance (f) 200 b) For your rocke's fligh, wrie he equaion relaing ime o heigh. h() = c) Wha is he independen variable (domain)? Wha is he dependen variable (range)? d) Once you have calculaed your iniial velociy, complee he able of values for he equaion. Be sure o include he daa for your launch and land poins h() h() e) Graph hese ordered pairs. f) How high did your rocke fly? Time (sec)

5 The Mah Projecs Journal Page 5 Mission o MArs: SECOND LAunch 6. a) When will he highes poin of he fligh occur? b) Use your fligh ime o calculae your iniial velociy, v 0. h() = v 0 + h 0 c) For your rocke's fligh, wrie he equaion relaing ime o heigh. h() = d) Once you have calculaed your iniial velociy, make a able of values for he equaion. Choose your domain in erms of 0.5 seconds inervals. (i.e. = 0.5, 1.0, 1.5, 2.0, 2.5). Be sure o include he daa for your launch and land poins. Disance (f) h() h() e) Graph hese ordered pairs f) How high did your rocke fly? Time (sec)

6 The Mah Projecs Journal Page 6 Mission o MArs: rocket AnAlysis Given he equaion of your second rocke: h() = 7. a) Wha is he verex of his equaion? (, ) b) Wha do he coordinaes of his poin represen in regards o he fligh of he rocke? c) Where on he graph is his poin found? 8. a) Wha are he roos of he equaion? (, ) & (, ) b) Wha do he roos represen in regards o he fligh of he rocke? c) Why is one of he roos negaive? d) Where on he graph are hese poins found?

7 The Mah Projecs Journal Page 7 Mission o MArs: rocket AnAlysis (coninued) 9. a) Wha is he y-inercep of his equaion? (, ) b) Wha do he coordinaes of his poin represen in regards o he fligh of he rocke? c) Where on he graph is his poin found? 10. How high will he rocke be afer 2 seconds? f 11. a) When will he rocke be 10 fee off he ground? sec b) Why are here wo soluions? 12. a) When will he rocke be 500 fee off he ground? sec b) Why are here no real soluions?