Tennessee Department of Education

Transcription

1 Teessee Departmet of Educatio Task: Comparig Shapes Geometry O a piece of graph paper with a coordiate plae, draw three o-colliear poits ad label them A, B, C. (Do ot use the origi as oe of your poits.) Coect these poits to make a triagle. For each poit, take half of the x ad y-coordiates ad label these ew poits A, B, C. Coect these poits to make aother triagle. 1. Compare the distace from the origi to poit A ad from the origi to poit A. Do the same for poits B ad B, ad for poits C ad C. Describe ay relatioships you otice.. Fid the perimeter of triagle ABC ad fid the perimeter of triagle A B C. Describe ay relatioships that you otice. 3. Suppose you repeated the directios, but you took a third of the x ad y coordiates. Make a cojecture about what would happe to the relatioships you oticed i parts 1 ad. 4. Suppose you repeated the directios, but used a differet shape (e.g. quadrilateral, petago, hexago). Make a cojecture about what would happe to the relatioships you oticed i parts 1 ad. 5. Verify your cojectures for umbers 3 ad 4. Extesio: If studets kow how to fid the area of o-right triagles, iclude this after part. Studets will compare the area of triagle A B C with the area of triagle ABC. Studet should the fiish the other parts of the task, makig cojectures ad provig them, icludig the areas. Teacher Notes: Studets will eed to kow how to fid distace betwee two poits i this task. Because studets ca place poits wherever they wat, they should see these relatioships will hold i geeral for ay ratio ad ay shape placed aywhere i the plae. This task will provide a good foudatio for studyig similarity ad for lookig at similarity through dilatios from a poit. Commo Core State Stadards for Mathematical Cotet Commo Core State Stadards for Mathematical Practice G-GPE.B.6 Fid the poit o a directed lie segmet betwee two give poits that partitios the segmet i a give ratio. G-GPE.B.7 Use coordiates to compute perimeters of polygos ad areas of triagles ad rectagles, e.g., usig the distace formula. 1. Make sese of problems ad persevere i solvig them.. Reaso abstractly ad quatitatively. 3. Costruct viable argumets ad critique the reasoig of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Atted to precisio. 7. Look for ad make use of structure. 8. Look for ad express regularity i repeated reasoig.

2 Essetial Uderstadigs The perimeter of figures whose side legths are i a :m ratio will also be i a :m ratio because additio preserves this ratio. Behid every proof is a proof idea. Empirical verificatio is a importat part of the process of provig, but it ca ever, by itself, costitute a proof. Geometry uses a wide variety of kids of proofs. Explore Phase Possible Solutio Paths Assessig ad Examiig Relatioships Studets ca place poits A, B, C aywhere i the plae. For example: A(-, 6) B(1, 7) A (-1, 3) C (5, -1) B (6, 3.5) C(10, -) Tell me why you placed your poits there. How did you kow where to place poits A, B, C? How ca you fid the distace betwee two poits? How do you calculate perimeter? The distace from the origi to A is d = ( ) + 6 = = 40 = 10 The distace from the origi to A is d = ( 1) + 3 = 1+ 9 = 10 The distace from the origi to A is half the distace from the origi to A. The steps would be similar for poits B ad B, ad for poits C ad C.

3 Depedig o where studets placed their poits, they might be able to cout without usig the distace formula, but ot beig able to use the origi should require them to use it somewhere. Studets should otice that the perimeter of the triagle with the prime poits is half of the perimeter of the origial triagle. Makig Cojectures Studets should make a cojecture that if you divide the x ad y- coordiates by ay umber, that umber will defie the ratio from the origi to the ew poit ad the origi to the origial poit. Similarly, studets should cojecture the relatioships will be maitaied with ay ratio ad the perimeter of ay polygo. Provig Cojectures Ratios from the Origi Suppose studets divide the x ad y-coordiates by : r s Let the coordiates for A be (r, s) ad the coordiates for a be,. The distace from the origi to A is d = r + s The distace from the origi to A is s r + s 1 d = + = = r + r The distace from the origi to A is 1 the distace from the origi to A. (Because this verificatio is doe i geeral, it will hold for ay poits B ad C. If studets use specific poits, ask how they ca geeralize their thikig to accout for all poits). s Tell me about your cojectures. Do you thik your cojecture will hold by dividig the coordiates by ay umber? Why or why ot? How could you show this for ay value? For ay polygo? I see you picked some ew poits ad tried a differet situatio. Tell me how you are thikig about verifyig your cojectures. Ca you verify your cojecture will be true for the ratio of everyoe i the class? Ca you verify your cojecture for ay polygo? How would you do that? How ca you express your reasoig? Would it be easier to explai i words, with algebra, or some other way? Ratio of Perimeters Let the coordiates for A be (r, s) ad the coordiates for B be (p, q). The distace from A to B is d = ( p r) + ( q s)

4 The distace from A to B is d = p r q s + = ( p r) + ( q s) = 1 ( p r) + ( q s) Studets ca repeat for the distace from A to C, B to C ad A to C, B to C. Whe summig for the perimeter, the 1 will factor out for triagle A B C. These ideas hold for ay polygo. Possible Studet Miscoceptios Studets may coect A to A, B to B, C to C ad ot have a accurate picture for the task. Studets may have a hard time thikig about how to verify their cojectures geerally. Studets may prove a specific case. Etry/Extesios If studets ca t get started. If studets fiish early. Let s read the problem together. What is it that you eed to do first? If you are to divide the coordiates by two, what kids of umbers might you choose that would make this problem a little easier. Tell me how you thik you should use these poits to make two triagles. Assessig Tell me about how you made this cojecture. What do you thik is happeig? What do you thik is happeig with your classmates relatioships? Advacig I see you are verifyig your cojecture for a quadrilateral you made. How ca you verify your cojecture for ay quadrilateral? Assessig ad Let s read the problem together. What is it that you eed to do first? Do you remember how to use the distace formula? Show me how to fid the distace betwee these two poits. What did you otice about the relatioships i geeral? What kid of proof did you use?

5 What do you thik will happe with the areas of the triagles? Why? Ca you prove this? Discuss/Aalyze Whole Group Questios What relatioships did you otice for the ratio from the origi to the prime poits ad the origial poits? Why do you thik this is the case? What about the ratio of the perimeters? What ca we say if these relatioships were true for everyoe s picture i the class, but everyoe graphed three differet poits to start with? How does this make you thik about a idea for a proof? Let s talk about your cojectures. What did you thik would happe by dividig the coordiates by a umber differet tha? Did ayoe try 3 or 4? What effect would this have o the differet ratios? I oticed that some of you verified your cojecture with a differet case. Remember we just said that everyoe i the class could have a differet picture but oticed all of the same relatioships. We wat to be able to verify a cojecture that will work for ay case. How ca we do that? Why do you thik it is importat to prove somethig that will work for ay case rather tha a specific oe?

6 Comparig Shapes Task Name: O a piece of graph paper, draw three o-colliear poits o a coordiate plae, ad label them A, B, C. (Do ot use the origi as oe of your poits.) Coect these poits to make a triagle. For each poit, take half of the x ad y-coordiates ad label these ew poits A, B, C. Coect these poits to make aother triagle. 1. Compare the distace from the origi to poit A ad from the origi to poit A. Do the same for poits B ad B, ad for poits C ad C. Describe ay relatioships you otice.. Fid the perimeter of triagle ABC ad fid the perimeter of triagle A B C. Describe ay relatioships that you otice. 3. Suppose you repeated the directios, but you took a third of the x ad y coordiates. Make a cojecture about what would happe to the relatioships you oticed i parts 1 ad.

7 4. Suppose you repeated the directios, but used a differet shape (e.g. quadrilateral, petago, hexago). Make a cojecture about what would happe to the relatioships you oticed i parts 1 ad. 5. Verify your cojectures for umbers 3 ad 4.